AND, OR and NOT Bernd Schr oder logo1 Bernd Schr oder Louisiana - - PowerPoint PPT Presentation

▶
and or and not
SMART_READER_LITE
LIVE PREVIEW

AND, OR and NOT Bernd Schr oder logo1 Bernd Schr oder Louisiana - - PowerPoint PPT Presentation

Apr 11, 2024 473 likes •2.5k views

Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules AND, OR and NOT Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-1
SLIDE 1

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

AND, OR and NOT

Bernd Schr¨

  • der

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-2
SLIDE 2

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logical Connectives in Daily Life

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-3
SLIDE 3

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logical Connectives in Daily Life

  • 1. The word “and” means that the statements on both sides

must be true

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-4
SLIDE 4

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logical Connectives in Daily Life

  • 1. The word “and” means that the statements on both sides

must be true: “We went out for dinner and we watched a movie.”

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-5
SLIDE 5

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logical Connectives in Daily Life

  • 1. The word “and” means that the statements on both sides

must be true: “We went out for dinner and we watched a movie.”

  • 2. The word “not” means that a statement is not true

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-6
SLIDE 6

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logical Connectives in Daily Life

  • 1. The word “and” means that the statements on both sides

must be true: “We went out for dinner and we watched a movie.”

  • 2. The word “not” means that a statement is not true: “We did

not make plans for next weekend.”

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-7
SLIDE 7

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logical Connectives in Daily Life

  • 1. The word “and” means that the statements on both sides

must be true: “We went out for dinner and we watched a movie.”

  • 2. The word “not” means that a statement is not true: “We did

not make plans for next weekend.”

  • 3. The word “or” means that at least one of the statements on

either side must be true

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-8
SLIDE 8

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logical Connectives in Daily Life

  • 1. The word “and” means that the statements on both sides

must be true: “We went out for dinner and we watched a movie.”

  • 2. The word “not” means that a statement is not true: “We did

not make plans for next weekend.”

  • 3. The word “or” means that at least one of the statements on

either side must be true: “They went out for dinner or they watched a movie.”

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-9
SLIDE 9

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logical Connectives in Daily Life

  • 1. The word “and” means that the statements on both sides

must be true: “We went out for dinner and we watched a movie.”

  • 2. The word “not” means that a statement is not true: “We did

not make plans for next weekend.”

  • 3. The word “or” means that at least one of the statements on

either side must be true: “They went out for dinner or they watched a movie.” We must be careful with “or”

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-10
SLIDE 10

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logical Connectives in Daily Life

  • 1. The word “and” means that the statements on both sides

must be true: “We went out for dinner and we watched a movie.”

  • 2. The word “not” means that a statement is not true: “We did

not make plans for next weekend.”

  • 3. The word “or” means that at least one of the statements on

either side must be true: “They went out for dinner or they watched a movie.” We must be careful with “or”: In everyday language the word “or” usually implies exclusivity, that is, it is assumed that both statements are not (or cannot be) true.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-11
SLIDE 11

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logical Connectives in Daily Life

  • 1. The word “and” means that the statements on both sides

must be true: “We went out for dinner and we watched a movie.”

  • 2. The word “not” means that a statement is not true: “We did

not make plans for next weekend.”

  • 3. The word “or” means that at least one of the statements on

either side must be true: “They went out for dinner or they watched a movie.” We must be careful with “or”: In everyday language the word “or” usually implies exclusivity, that is, it is assumed that both statements are not (or cannot be) true. In mathematics, “or” is inclusive

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-12
SLIDE 12

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logical Connectives in Daily Life

  • 1. The word “and” means that the statements on both sides

must be true: “We went out for dinner and we watched a movie.”

  • 2. The word “not” means that a statement is not true: “We did

not make plans for next weekend.”

  • 3. The word “or” means that at least one of the statements on

either side must be true: “They went out for dinner or they watched a movie.” We must be careful with “or”: In everyday language the word “or” usually implies exclusivity, that is, it is assumed that both statements are not (or cannot be) true. In mathematics, “or” is inclusive, that is, both statements can be true.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-13
SLIDE 13

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logical Connectives in Daily Life

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-14
SLIDE 14

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logical Connectives in Daily Life

  • 1. Let’s say we want to fix dinner

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-15
SLIDE 15

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logical Connectives in Daily Life

  • 1. Let’s say we want to fix dinner, a beef dish.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-16
SLIDE 16

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logical Connectives in Daily Life

  • 1. Let’s say we want to fix dinner, a beef dish.
  • 2. Use a search engine to find a recipe.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-17
SLIDE 17

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logical Connectives in Daily Life

  • 1. Let’s say we want to fix dinner, a beef dish.
  • 2. Use a search engine to find a recipe.

Search term: “beef”.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-18
SLIDE 18

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logical Connectives in Daily Life

  • 1. Let’s say we want to fix dinner, a beef dish.
  • 2. Use a search engine to find a recipe.

Search term: “beef”.

  • 3. Too many results. Say we want noodles with it.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-19
SLIDE 19

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logical Connectives in Daily Life

  • 1. Let’s say we want to fix dinner, a beef dish.
  • 2. Use a search engine to find a recipe.

Search term: “beef”.

  • 3. Too many results. Say we want noodles with it.

Search term: “beef and noodles”.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-20
SLIDE 20

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logical Connectives in Daily Life

  • 1. Let’s say we want to fix dinner, a beef dish.
  • 2. Use a search engine to find a recipe.

Search term: “beef”.

  • 3. Too many results. Say we want noodles with it.

Search term: “beef and noodles”.

  • 4. Still a lot ... maybe with cream sauce.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-21
SLIDE 21

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logical Connectives in Daily Life

  • 1. Let’s say we want to fix dinner, a beef dish.
  • 2. Use a search engine to find a recipe.

Search term: “beef”.

  • 3. Too many results. Say we want noodles with it.

Search term: “beef and noodles”.

  • 4. Still a lot ... maybe with cream sauce.

Search term: “beef and noodles and cream”.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-22
SLIDE 22

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for AND

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-23
SLIDE 23

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for AND

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-24
SLIDE 24

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for AND

p

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-25
SLIDE 25

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for AND

p q

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-26
SLIDE 26

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for AND

p q p∧q

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-27
SLIDE 27

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for AND

p q p∧q FALSE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-28
SLIDE 28

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for AND

p q p∧q FALSE FALSE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-29
SLIDE 29

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for AND

p q p∧q FALSE FALSE FALSE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-30
SLIDE 30

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for AND

p q p∧q FALSE FALSE FALSE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-31
SLIDE 31

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for AND

p q p∧q FALSE FALSE FALSE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-32
SLIDE 32

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for AND

p q p∧q FALSE FALSE FALSE TRUE TRUE FALSE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-33
SLIDE 33

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for AND

p q p∧q FALSE FALSE FALSE TRUE TRUE FALSE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-34
SLIDE 34

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for AND

p q p∧q FALSE FALSE FALSE TRUE TRUE FALSE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-35
SLIDE 35

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for AND

p q p∧q FALSE FALSE FALSE FALSE TRUE TRUE FALSE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-36
SLIDE 36

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for AND

p q p∧q FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-37
SLIDE 37

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for AND

p q p∧q FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-38
SLIDE 38

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for AND

p q p∧q FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-39
SLIDE 39

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for OR

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-40
SLIDE 40

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for OR

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-41
SLIDE 41

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for OR

p

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-42
SLIDE 42

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for OR

p q

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-43
SLIDE 43

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for OR

p q p∨q

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-44
SLIDE 44

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for OR

p q p∨q FALSE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-45
SLIDE 45

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for OR

p q p∨q FALSE FALSE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-46
SLIDE 46

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for OR

p q p∨q FALSE FALSE FALSE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-47
SLIDE 47

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for OR

p q p∨q FALSE FALSE FALSE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-48
SLIDE 48

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for OR

p q p∨q FALSE FALSE FALSE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-49
SLIDE 49

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for OR

p q p∨q FALSE FALSE FALSE TRUE TRUE FALSE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-50
SLIDE 50

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for OR

p q p∨q FALSE FALSE FALSE TRUE TRUE FALSE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-51
SLIDE 51

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for OR

p q p∨q FALSE FALSE FALSE TRUE TRUE FALSE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-52
SLIDE 52

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for OR

p q p∨q FALSE FALSE FALSE FALSE TRUE TRUE FALSE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-53
SLIDE 53

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for OR

p q p∨q FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-54
SLIDE 54

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for OR

p q p∨q FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-55
SLIDE 55

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for OR

p q p∨q FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-56
SLIDE 56

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for NOT

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-57
SLIDE 57

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for NOT

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-58
SLIDE 58

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for NOT

p

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-59
SLIDE 59

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for NOT

p ¬p

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-60
SLIDE 60

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for NOT

p ¬p FALSE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-61
SLIDE 61

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for NOT

p ¬p FALSE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-62
SLIDE 62

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for NOT

p ¬p FALSE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-63
SLIDE 63

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

The Truth Table for NOT

p ¬p FALSE TRUE TRUE FALSE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-64
SLIDE 64

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Visualization With Switches

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-65
SLIDE 65

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Visualization With Switches

  • 1. Every switch represents a primitive proposition.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-66
SLIDE 66

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Visualization With Switches

  • 1. Every switch represents a primitive proposition.
  • 2. Open switch = FALSE, closed switch = TRUE.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-67
SLIDE 67

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Visualization With Switches

  • 1. Every switch represents a primitive proposition.
  • 2. Open switch = FALSE, closed switch = TRUE.
  • 3. Compound statement represented by the circuit.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-68
SLIDE 68

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Visualization With Switches

  • 1. Every switch represents a primitive proposition.
  • 2. Open switch = FALSE, closed switch = TRUE.
  • 3. Compound statement represented by the circuit.
  • 4. Conducting connection from A to B: compound statement

TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-69
SLIDE 69

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Visualization With Switches

  • 1. Every switch represents a primitive proposition.
  • 2. Open switch = FALSE, closed switch = TRUE.
  • 3. Compound statement represented by the circuit.
  • 4. Conducting connection from A to B: compound statement

TRUE, otherwise false.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-70
SLIDE 70

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Switch Diagram for OR

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-71
SLIDE 71

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Switch Diagram for OR

  • ❜

❜ r r r r p q A B Disjunction r r

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-72
SLIDE 72

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Switch Diagram for AND

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-73
SLIDE 73

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Switch Diagram for AND

  • ❜

❜ r r r r p q A B Conjunction

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-74
SLIDE 74

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logic Gates

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-75
SLIDE 75

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logic Gates

  • 1. Switch diagrams are for explanations only.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-76
SLIDE 76

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logic Gates

  • 1. Switch diagrams are for explanations only.
  • 2. But logic circuits are also used in practical applications.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-77
SLIDE 77

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logic Gates

  • 1. Switch diagrams are for explanations only.
  • 2. But logic circuits are also used in practical applications.
  • 3. There are three kinds of gates.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-78
SLIDE 78

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logic Gates

  • 1. Switch diagrams are for explanations only.
  • 2. But logic circuits are also used in practical applications.
  • 3. There are three kinds of gates.

AND-gate

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-79
SLIDE 79

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logic Gates

  • 1. Switch diagrams are for explanations only.
  • 2. But logic circuits are also used in practical applications.
  • 3. There are three kinds of gates.

AND-gate OR-gate

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-80
SLIDE 80

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Logic Gates

  • 1. Switch diagrams are for explanations only.
  • 2. But logic circuits are also used in practical applications.
  • 3. There are three kinds of gates.

AND-gate OR-gate NOT-gate (inverter) ✟✟ ✟ ❍ ❍ ❍ r

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-81
SLIDE 81

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

All Logic Circuits are Constructed from AND-, OR- and NOT-Gates

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-82
SLIDE 82

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

All Logic Circuits are Constructed from AND-, OR- and NOT-Gates

... and every computer is essentially a logic circuit.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-83
SLIDE 83

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

All Logic Circuits are Constructed from AND-, OR- and NOT-Gates

... and every computer is essentially a logic circuit. How can that be?

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-84
SLIDE 84

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

All Logic Circuits are Constructed from AND-, OR- and NOT-Gates

... and every computer is essentially a logic circuit. How can that be? Circuit

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-85
SLIDE 85

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

All Logic Circuits are Constructed from AND-, OR- and NOT-Gates

... and every computer is essentially a logic circuit. How can that be? Circuit

inputs

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-86
SLIDE 86

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

All Logic Circuits are Constructed from AND-, OR- and NOT-Gates

... and every computer is essentially a logic circuit. How can that be? Circuit

p inputs

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-87
SLIDE 87

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

All Logic Circuits are Constructed from AND-, OR- and NOT-Gates

... and every computer is essentially a logic circuit. How can that be? Circuit

p q inputs

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-88
SLIDE 88

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

All Logic Circuits are Constructed from AND-, OR- and NOT-Gates

... and every computer is essentially a logic circuit. How can that be? Circuit

p q r inputs

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-89
SLIDE 89

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

All Logic Circuits are Constructed from AND-, OR- and NOT-Gates

... and every computer is essentially a logic circuit. How can that be? Circuit

p q r inputs

  • utputs

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-90
SLIDE 90

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

All Logic Circuits are Constructed from AND-, OR- and NOT-Gates

... and every computer is essentially a logic circuit. How can that be? Circuit

p f1(p,q,r) q r inputs

  • utputs

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-91
SLIDE 91

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

All Logic Circuits are Constructed from AND-, OR- and NOT-Gates

... and every computer is essentially a logic circuit. How can that be? Circuit

p f1(p,q,r) f2(p,q,r) q r inputs

  • utputs

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-92
SLIDE 92

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

All Logic Circuits are Constructed from AND-, OR- and NOT-Gates

... and every computer is essentially a logic circuit. How can that be? Circuit

p f1(p,q,r) f2(p,q,r) f3(p,q,r) q r inputs

  • utputs

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-93
SLIDE 93

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

All Logic Circuits are Constructed from AND-, OR- and NOT-Gates

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-94
SLIDE 94

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

All Logic Circuits are Constructed from AND-, OR- and NOT-Gates

If we have two entries u and v,

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-95
SLIDE 95

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

All Logic Circuits are Constructed from AND-, OR- and NOT-Gates

If we have two entries u and v,

  • 1. ¬u∧¬v is true iff u is false and v is false.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-96
SLIDE 96

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

All Logic Circuits are Constructed from AND-, OR- and NOT-Gates

If we have two entries u and v,

  • 1. ¬u∧¬v is true iff u is false and v is false.
  • 2. ¬u∧v is true iff u is false and v is true.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-97
SLIDE 97

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

All Logic Circuits are Constructed from AND-, OR- and NOT-Gates

If we have two entries u and v,

  • 1. ¬u∧¬v is true iff u is false and v is false.
  • 2. ¬u∧v is true iff u is false and v is true.
  • 3. u∧¬v is true iff u is true and v is false.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-98
SLIDE 98

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

All Logic Circuits are Constructed from AND-, OR- and NOT-Gates

If we have two entries u and v,

  • 1. ¬u∧¬v is true iff u is false and v is false.
  • 2. ¬u∧v is true iff u is false and v is true.
  • 3. u∧¬v is true iff u is true and v is false.
  • 4. u∧v is true iff u is true and v is true.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-99
SLIDE 99

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

All Logic Circuits are Constructed from AND-, OR- and NOT-Gates

If we have two entries u and v,

  • 1. ¬u∧¬v is true iff u is false and v is false.
  • 2. ¬u∧v is true iff u is false and v is true.
  • 3. u∧¬v is true iff u is true and v is false.
  • 4. u∧v is true iff u is true and v is true.

Similar constructions can be done for more than 2 entries: Put ¬ where you need a FALSE for the output to be true.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-100
SLIDE 100

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

All Logic Circuits are Constructed from AND-, OR- and NOT-Gates

If we have two entries u and v,

  • 1. ¬u∧¬v is true iff u is false and v is false.
  • 2. ¬u∧v is true iff u is false and v is true.
  • 3. u∧¬v is true iff u is true and v is false.
  • 4. u∧v is true iff u is true and v is true.

Similar constructions can be done for more than 2 entries: Put ¬ where you need a FALSE for the output to be true. For every input that gives a true output, construct a statement that is true exactly for that input.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-101
SLIDE 101

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

All Logic Circuits are Constructed from AND-, OR- and NOT-Gates

If we have two entries u and v,

  • 1. ¬u∧¬v is true iff u is false and v is false.
  • 2. ¬u∧v is true iff u is false and v is true.
  • 3. u∧¬v is true iff u is true and v is false.
  • 4. u∧v is true iff u is true and v is true.

Similar constructions can be done for more than 2 entries: Put ¬ where you need a FALSE for the output to be true. For every input that gives a true output, construct a statement that is true exactly for that input. Connect all these statements with an OR.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-102
SLIDE 102

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

All Logic Circuits are Constructed from AND-, OR- and NOT-Gates

If we have two entries u and v,

  • 1. ¬u∧¬v is true iff u is false and v is false.
  • 2. ¬u∧v is true iff u is false and v is true.
  • 3. u∧¬v is true iff u is true and v is false.
  • 4. u∧v is true iff u is true and v is true.

Similar constructions can be done for more than 2 entries: Put ¬ where you need a FALSE for the output to be true. For every input that gives a true output, construct a statement that is true exactly for that input. Connect all these statements with an OR. Now check out the proof in the book once more, because we need to get used to the requisite precision.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-103
SLIDE 103

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

A Sample Logic Circuit

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-104
SLIDE 104

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

A Sample Logic Circuit

Specifications:

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-105
SLIDE 105

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

A Sample Logic Circuit

Specifications:

  • 1. 3 inputs: p, q, r.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-106
SLIDE 106

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

A Sample Logic Circuit

Specifications:

  • 1. 3 inputs: p, q, r.
  • 2. When p is false, the output will be false.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-107
SLIDE 107

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

A Sample Logic Circuit

Specifications:

  • 1. 3 inputs: p, q, r.
  • 2. When p is false, the output will be false.
  • 3. When p is true, the output will be true iff q and r have

different values.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-108
SLIDE 108

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

A Sample Logic Circuit

Specifications:

  • 1. 3 inputs: p, q, r.
  • 2. When p is false, the output will be false.
  • 3. When p is true, the output will be true iff q and r have

different values. That means there are only two input combinations that give a TRUE output.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-109
SLIDE 109

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

A Sample Logic Circuit

Specifications:

  • 1. 3 inputs: p, q, r.
  • 2. When p is false, the output will be false.
  • 3. When p is true, the output will be true iff q and r have

different values. That means there are only two input combinations that give a TRUE output. In the order (p,q,r), these inputs are (TRUE,TRUE,FALSE) and (TRUE,FALSE,TRUE)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-110
SLIDE 110

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

A Sample Logic Circuit

Specifications:

  • 1. 3 inputs: p, q, r.
  • 2. When p is false, the output will be false.
  • 3. When p is true, the output will be true iff q and r have

different values. That means there are only two input combinations that give a TRUE output. In the order (p,q,r), these inputs are (TRUE,TRUE,FALSE) and (TRUE,FALSE,TRUE) Statement:

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-111
SLIDE 111

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

A Sample Logic Circuit

Specifications:

  • 1. 3 inputs: p, q, r.
  • 2. When p is false, the output will be false.
  • 3. When p is true, the output will be true iff q and r have

different values. That means there are only two input combinations that give a TRUE output. In the order (p,q,r), these inputs are (TRUE,TRUE,FALSE) and (TRUE,FALSE,TRUE) Statement: (p∧q∧¬r)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-112
SLIDE 112

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

A Sample Logic Circuit

Specifications:

  • 1. 3 inputs: p, q, r.
  • 2. When p is false, the output will be false.
  • 3. When p is true, the output will be true iff q and r have

different values. That means there are only two input combinations that give a TRUE output. In the order (p,q,r), these inputs are (TRUE,TRUE,FALSE) and (TRUE,FALSE,TRUE) Statement: (p∧q∧¬r)∨(p∧¬q∧r).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-113
SLIDE 113

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

A Sample Logic Circuit

Specifications:

  • 1. 3 inputs: p, q, r.
  • 2. When p is false, the output will be false.
  • 3. When p is true, the output will be true iff q and r have

different values. That means there are only two input combinations that give a TRUE output. In the order (p,q,r), these inputs are (TRUE,TRUE,FALSE) and (TRUE,FALSE,TRUE) Statement: (p∧q∧¬r)∨(p∧¬q∧r). Circuit uses 5 AND- or OR-gates.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-114
SLIDE 114

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

(p∧q∧¬r)∨(p∧¬q∧r)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-115
SLIDE 115

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

(p∧q∧¬r)∨(p∧¬q∧r)

p r q Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-116
SLIDE 116

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

(p∧q∧¬r)∨(p∧¬q∧r)

p r q Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-117
SLIDE 117

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

(p∧q∧¬r)∨(p∧¬q∧r)

p r q Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-118
SLIDE 118

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

❍❍ ❍ ✟ ✟ ✟ r

(p∧q∧¬r)∨(p∧¬q∧r)

p r q Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-119
SLIDE 119

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

❍❍ ❍ ✟ ✟ ✟ r

(p∧q∧¬r)∨(p∧¬q∧r)

p r q Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-120
SLIDE 120

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

❍❍ ❍ ✟ ✟ ✟ r

(p∧q∧¬r)∨(p∧¬q∧r)

r

p r q Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-121
SLIDE 121

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

❍❍ ❍ ✟ ✟ ✟ r

(p∧q∧¬r)∨(p∧¬q∧r)

r

p r q Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-122
SLIDE 122

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

❍❍ ❍ ❍❍ ❍ ✟ ✟ ✟ ✟ ✟ ✟ r r

(p∧q∧¬r)∨(p∧¬q∧r)

r r

p r q Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-123
SLIDE 123

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

❍❍ ❍ ❍❍ ❍ ✟ ✟ ✟ ✟ ✟ ✟ r r

(p∧q∧¬r)∨(p∧¬q∧r)

r r r

p r q Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-124
SLIDE 124

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

❍❍ ❍ ❍❍ ❍ ✟ ✟ ✟ ✟ ✟ ✟ r r

(p∧q∧¬r)∨(p∧¬q∧r)

r r r

p r q Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-125
SLIDE 125

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

❍❍ ❍ ❍❍ ❍ ✟ ✟ ✟ ✟ ✟ ✟ r r

(p∧q∧¬r)∨(p∧¬q∧r)

r r r

p r q Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-126
SLIDE 126

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Why Would We Want To Do Algebra With Logical Connectives?

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-127
SLIDE 127

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Why Would We Want To Do Algebra With Logical Connectives?

  • 1. Compound statements can become needlessly complicated.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-128
SLIDE 128

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Why Would We Want To Do Algebra With Logical Connectives?

  • 1. Compound statements can become needlessly complicated.
  • 2. In circuit design, every conjunction and disjunction is

implemented with a gate.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-129
SLIDE 129

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Why Would We Want To Do Algebra With Logical Connectives?

  • 1. Compound statements can become needlessly complicated.
  • 2. In circuit design, every conjunction and disjunction is

implemented with a gate.

  • 3. Actually, the real implementation is a bit more

complicated, but it ultimately involves algebra, too. (All logic circuits are based on NAND-gates.)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-130
SLIDE 130

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Boolean Algebra, Part I

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-131
SLIDE 131

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Boolean Algebra, Part I

  • 1. p∧q = q∧p (AND is commutative)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-132
SLIDE 132

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Boolean Algebra, Part I

  • 1. p∧q = q∧p (AND is commutative)
  • 2. p∨q = q∨p (OR is commutative)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-133
SLIDE 133

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Boolean Algebra, Part I

  • 1. p∧q = q∧p (AND is commutative)
  • 2. p∨q = q∨p (OR is commutative)
  • 3. (p∧q)∧r = p∧(q∧r) (AND is associative)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-134
SLIDE 134

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Boolean Algebra, Part I

  • 1. p∧q = q∧p (AND is commutative)
  • 2. p∨q = q∨p (OR is commutative)
  • 3. (p∧q)∧r = p∧(q∧r) (AND is associative)
  • 4. (p∨q)∨r = p∨(q∨r) (OR is associative)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-135
SLIDE 135

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Boolean Algebra, Part I

  • 1. p∧q = q∧p (AND is commutative)
  • 2. p∨q = q∨p (OR is commutative)
  • 3. (p∧q)∧r = p∧(q∧r) (AND is associative)
  • 4. (p∨q)∨r = p∨(q∨r) (OR is associative)
  • 5. p∧(q∨r) = (p∧q)∨(p∧r) (AND is distributive over

OR)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-136
SLIDE 136

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Boolean Algebra, Part I

  • 1. p∧q = q∧p (AND is commutative)
  • 2. p∨q = q∨p (OR is commutative)
  • 3. (p∧q)∧r = p∧(q∧r) (AND is associative)
  • 4. (p∨q)∨r = p∨(q∨r) (OR is associative)
  • 5. p∧(q∨r) = (p∧q)∨(p∧r) (AND is distributive over

OR)

  • 6. p∨(q∧r) = (p∨q)∧(p∨r) (OR is distributive over

AND)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-137
SLIDE 137

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-138
SLIDE 138

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r FALSE FALSE FALSE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-139
SLIDE 139

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r FALSE FALSE FALSE FALSE FALSE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-140
SLIDE 140

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-141
SLIDE 141

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-142
SLIDE 142

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE FALSE FALSE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-143
SLIDE 143

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE FALSE FALSE TRUE FALSE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-144
SLIDE 144

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-145
SLIDE 145

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-146
SLIDE 146

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-147
SLIDE 147

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p ∧(q∨r) FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-148
SLIDE 148

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q ∧(q∨r) FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-149
SLIDE 149

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r ∧(q∨r) FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-150
SLIDE 150

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-151
SLIDE 151

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-152
SLIDE 152

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-153
SLIDE 153

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE TRUE FALSE TRUE FALSE TRUE TRUE TRUE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-154
SLIDE 154

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE TRUE FALSE TRUE FALSE TRUE TRUE TRUE TRUE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-155
SLIDE 155

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE TRUE FALSE TRUE FALSE TRUE TRUE TRUE TRUE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-156
SLIDE 156

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE TRUE FALSE TRUE FALSE TRUE TRUE TRUE TRUE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-157
SLIDE 157

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE TRUE FALSE TRUE FALSE TRUE TRUE TRUE TRUE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-158
SLIDE 158

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE TRUE FALSE TRUE FALSE TRUE TRUE TRUE TRUE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-159
SLIDE 159

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE TRUE FALSE TRUE FALSE TRUE TRUE TRUE TRUE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-160
SLIDE 160

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE TRUE FALSE TRUE FALSE TRUE TRUE TRUE TRUE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-161
SLIDE 161

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE TRUE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-162
SLIDE 162

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE FALSE TRUE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-163
SLIDE 163

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE FALSE TRUE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-164
SLIDE 164

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE FALSE TRUE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-165
SLIDE 165

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE FALSE TRUE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-166
SLIDE 166

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE FALSE TRUE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-167
SLIDE 167

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE FALSE TRUE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-168
SLIDE 168

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE FALSE TRUE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-169
SLIDE 169

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE TRUE TRUE TRUE FALSE TRUE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-170
SLIDE 170

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE TRUE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-171
SLIDE 171

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-172
SLIDE 172

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-173
SLIDE 173

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-174
SLIDE 174

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-175
SLIDE 175

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-176
SLIDE 176

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-177
SLIDE 177

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-178
SLIDE 178

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-179
SLIDE 179

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-180
SLIDE 180

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-181
SLIDE 181

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-182
SLIDE 182

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-183
SLIDE 183

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-184
SLIDE 184

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-185
SLIDE 185

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-186
SLIDE 186

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-187
SLIDE 187

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-188
SLIDE 188

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-189
SLIDE 189

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-190
SLIDE 190

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

Proof of p∧(q∨r) = (p∧q)∨(p∧r)

p q r q∨r p p∧q p∧r (p∧q) ∧(q∨r) ∨(p∧r) FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-191
SLIDE 191

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

A Sample Computation

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-192
SLIDE 192

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

A Sample Computation

(p∧q∧¬r)∨(p∧¬q∧r)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-193
SLIDE 193

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

A Sample Computation

(p∧q∧¬r)∨(p∧¬q∧r) = p∧

  • (q∧¬r)∨(¬q∧r)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-194
SLIDE 194

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

A Sample Computation

(p∧q∧¬r)∨(p∧¬q∧r) = p∧

  • (q∧¬r)∨(¬q∧r)
  • =

p∧

  • q∨(¬q∧r)
  • ∧
  • ¬r ∨(¬q∧r)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-195
SLIDE 195

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

A Sample Computation

(p∧q∧¬r)∨(p∧¬q∧r) = p∧

  • (q∧¬r)∨(¬q∧r)
  • =

p∧

  • q∨(¬q∧r)
  • ∧
  • ¬r ∨(¬q∧r)
  • =

p∧

  • (q∨¬q)∧(q∨r)
  • ∧
  • (¬r ∨¬q)∧(¬r ∨r)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-196
SLIDE 196

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

A Sample Computation

(p∧q∧¬r)∨(p∧¬q∧r) = p∧

  • (q∧¬r)∨(¬q∧r)
  • =

p∧

  • q∨(¬q∧r)
  • ∧
  • ¬r ∨(¬q∧r)
  • =

p∧

  • (q∨¬q

=TRUE

)∧(q∨r)

  • ∧
  • (¬r ∨¬q)∧(¬r ∨r)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-197
SLIDE 197

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

A Sample Computation

(p∧q∧¬r)∨(p∧¬q∧r) = p∧

  • (q∧¬r)∨(¬q∧r)
  • =

p∧

  • q∨(¬q∧r)
  • ∧
  • ¬r ∨(¬q∧r)
  • =

p∧

  • (q∨¬q

=TRUE

)∧(q∨r)

  • ∧
  • (¬r ∨¬q)∧(¬r ∨r

=TRUE

)

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-198
SLIDE 198

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

A Sample Computation

(p∧q∧¬r)∨(p∧¬q∧r) = p∧

  • (q∧¬r)∨(¬q∧r)
  • =

p∧

  • q∨(¬q∧r)
  • ∧
  • ¬r ∨(¬q∧r)
  • =

p∧

  • (q∨¬q

=TRUE

)∧(q∨r)

  • ∧
  • (¬r ∨¬q)∧(¬r ∨r

=TRUE

)

  • =

p∧

  • (q∨r)∧(¬r ∨¬q)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-199
SLIDE 199

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

A Sample Computation

(p∧q∧¬r)∨(p∧¬q∧r) = p∧

  • (q∧¬r)∨(¬q∧r)
  • =

p∧

  • q∨(¬q∧r)
  • ∧
  • ¬r ∨(¬q∧r)
  • =

p∧

  • (q∨¬q

=TRUE

)∧(q∨r)

  • ∧
  • (¬r ∨¬q)∧(¬r ∨r

=TRUE

)

  • =

p∧

  • (q∨r)∧(¬r ∨¬q)
  • Simplified result needs one less AND/OR operation.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT

slide-200
SLIDE 200

logo1 Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules

A Sample Computation

(p∧q∧¬r)∨(p∧¬q∧r) = p∧

  • (q∧¬r)∨(¬q∧r)
  • =

p∧

  • q∨(¬q∧r)
  • ∧
  • ¬r ∨(¬q∧r)
  • =

p∧

  • (q∨¬q

=TRUE

)∧(q∨r)

  • ∧
  • (¬r ∨¬q)∧(¬r ∨r

=TRUE

)

  • =

p∧

  • (q∨r)∧(¬r ∨¬q)
  • Simplified result needs one less AND/OR operation.

(Sketching the circuit would be a good exercise.)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science AND, OR and NOT