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Propositional Logic Truth Tables Logical Equivalence Discrete Mathematics with Applications Chapter 2: The Logic of Compound Statements (Part 1) January 23, 2019 Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with

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Propositional Logic Truth Tables Logical Equivalence

Discrete Mathematics with Applications

Chapter 2: The Logic of Compound Statements (Part 1) January 23, 2019

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Statements

A statement (or proposition) is a sentence that is either true (T) or false (F), but not both.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Statements

A statement (or proposition) is a sentence that is either true (T) or false (F), but not both. Examples:

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Statements

A statement (or proposition) is a sentence that is either true (T) or false (F), but not both. Examples:

1 “The earth is flat.” – F

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Statements

A statement (or proposition) is a sentence that is either true (T) or false (F), but not both. Examples:

1 “The earth is flat.” – F 2 “March has 31 days.” – T

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Statements

A statement (or proposition) is a sentence that is either true (T) or false (F), but not both. Examples:

1 “The earth is flat.” – F 2 “March has 31 days.” – T 3 “Do you have the time?” – Not a statement (it’s a question)

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Statements

A statement (or proposition) is a sentence that is either true (T) or false (F), but not both. Examples:

1 “The earth is flat.” – F 2 “March has 31 days.” – T 3 “Do you have the time?” – Not a statement (it’s a question) 4 “Go to the store.” – Not a statement (it’s a command)

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Statements

A statement (or proposition) is a sentence that is either true (T) or false (F), but not both. Examples:

1 “The earth is flat.” – F 2 “March has 31 days.” – T 3 “Do you have the time?” – Not a statement (it’s a question) 4 “Go to the store.” – Not a statement (it’s a command)

Notation: Lower case letters are often used to represent statements.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Logical Connectives

Connectives are symbols that combine statements. Statements separated by connectives make a compound statement.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Logical Connectives

Connectives are symbols that combine statements. Statements separated by connectives make a compound statement. The negation of p is the statement “not p” or “it is not the case that p” and is denoted by ∼p.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Logical Connectives

Connectives are symbols that combine statements. Statements separated by connectives make a compound statement. The negation of p is the statement “not p” or “it is not the case that p” and is denoted by ∼p. The conjunction of the p and q is the statement “p and q” and is denoted by p ∧ q.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Logical Connectives

Connectives are symbols that combine statements. Statements separated by connectives make a compound statement. The negation of p is the statement “not p” or “it is not the case that p” and is denoted by ∼p. The conjunction of the p and q is the statement “p and q” and is denoted by p ∧ q. The disjunction of p and q is the statement “p or q” and is denoted by p ∨ q.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Inclusive Or vs. Exclusive Or

NOTE: The meaning of “or” here is inclusive, that is p ∨ q is true whenever p is true, or q is true, or both.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Inclusive Or vs. Exclusive Or

NOTE: The meaning of “or” here is inclusive, that is p ∨ q is true whenever p is true, or q is true, or both. e.g. “You will pass this course if you complete all of the homework or perform well on the exams.”

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Inclusive Or vs. Exclusive Or

NOTE: The meaning of “or” here is inclusive, that is p ∨ q is true whenever p is true, or q is true, or both. e.g. “You will pass this course if you complete all of the homework or perform well on the exams.” This is certainly a true statement, especially if you do both!

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Inclusive Or vs. Exclusive Or

NOTE: The meaning of “or” here is inclusive, that is p ∨ q is true whenever p is true, or q is true, or both. e.g. “You will pass this course if you complete all of the homework or perform well on the exams.” This is certainly a true statement, especially if you do both! When “or” is to be used in the exclusive sense, that is “p or q but not both” or “p or q but not both p and q,” then we write p ⊕ q or p XOR q.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Examples

p : “The earth is flat.”

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Examples

p : “The earth is flat.” q : “March has 31 days.”

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Examples

p : “The earth is flat.” q : “March has 31 days.” ∼p : “The earth is not flat.”

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Examples

p : “The earth is flat.” q : “March has 31 days.” ∼p : “The earth is not flat.” p ∧ q : “The earth is flat and March has 31 days.”

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Examples

p : “The earth is flat.” q : “March has 31 days.” ∼p : “The earth is not flat.” p ∧ q : “The earth is flat and March has 31 days.” p ∨ q : “The earth is flat or March has 31 days.”

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Order of Operations

∼ has the highest precedence, then ∧, then ∨.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Order of Operations

∼ has the highest precedence, then ∧, then ∨. ∧ and ∨ associate to the left – group two statements from the left.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Order of Operations

∼ has the highest precedence, then ∧, then ∨. ∧ and ∨ associate to the left – group two statements from the left. Parentheses can be used to indicate the order in which connectives should be taken. (It is a good habit to use parentheses to make compound statements more understandable.)

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Order of Operations

∼ has the highest precedence, then ∧, then ∨. ∧ and ∨ associate to the left – group two statements from the left. Parentheses can be used to indicate the order in which connectives should be taken. (It is a good habit to use parentheses to make compound statements more understandable.) e.g. p ∧ ∼q ∧ r means (p ∧ (∼q)) ∧ r

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Order of Operations

∼ has the highest precedence, then ∧, then ∨. ∧ and ∨ associate to the left – group two statements from the left. Parentheses can be used to indicate the order in which connectives should be taken. (It is a good habit to use parentheses to make compound statements more understandable.) e.g. p ∧ ∼q ∧ r means (p ∧ (∼q)) ∧ r p ∨ ∼q ∧ r means p ∨ ((∼q) ∧ r)

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Translate from English to Symbols

Write the following sentences symbolically, letting h represent “It is hot” and letting s represent “It is sunny.”

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Translate from English to Symbols

Write the following sentences symbolically, letting h represent “It is hot” and letting s represent “It is sunny.”

1 It is not hot, but it is sunny.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Translate from English to Symbols

Write the following sentences symbolically, letting h represent “It is hot” and letting s represent “It is sunny.”

1 It is not hot, but it is sunny. 2 It is neither hot nor sunny.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Translate from English to Symbols

Write the following sentences symbolically, letting h represent “It is hot” and letting s represent “It is sunny.”

1 It is not hot, but it is sunny. 2 It is neither hot nor sunny.

Answers

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Translate from English to Symbols

Write the following sentences symbolically, letting h represent “It is hot” and letting s represent “It is sunny.”

1 It is not hot, but it is sunny. 2 It is neither hot nor sunny.

Answers

1 ∼h ∧ s

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Translate from English to Symbols

Write the following sentences symbolically, letting h represent “It is hot” and letting s represent “It is sunny.”

1 It is not hot, but it is sunny. 2 It is neither hot nor sunny.

Answers

1 ∼h ∧ s 2 ∼h ∧ ∼s

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Logically Dealing with Inequalities

Let p, q, and r symbolize x > 0, x < 3, and x = 3,

respectively. Write the following inequalities symbolically.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Logically Dealing with Inequalities

Let p, q, and r symbolize x > 0, x < 3, and x = 3,

respectively. Write the following inequalities symbolically.

1 x ≤ 3

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Logically Dealing with Inequalities

Let p, q, and r symbolize x > 0, x < 3, and x = 3,

respectively. Write the following inequalities symbolically.

1 x ≤ 3 2 0 < x < 3

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Logically Dealing with Inequalities

Let p, q, and r symbolize x > 0, x < 3, and x = 3,

respectively. Write the following inequalities symbolically.

1 x ≤ 3 2 0 < x < 3 3 0 < x ≤ 3

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Logically Dealing with Inequalities

Let p, q, and r symbolize x > 0, x < 3, and x = 3,

respectively. Write the following inequalities symbolically.

1 x ≤ 3 2 0 < x < 3 3 0 < x ≤ 3

Answers

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Logically Dealing with Inequalities

Let p, q, and r symbolize x > 0, x < 3, and x = 3,

respectively. Write the following inequalities symbolically.

1 x ≤ 3 2 0 < x < 3 3 0 < x ≤ 3

Answers

1 x ≤ 3 means that x is less than or equal to 3, so we have q ∨ r.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Logically Dealing with Inequalities

Let p, q, and r symbolize x > 0, x < 3, and x = 3,

respectively. Write the following inequalities symbolically.

1 x ≤ 3 2 0 < x < 3 3 0 < x ≤ 3

Answers

1 x ≤ 3 means that x is less than or equal to 3, so we have q ∨ r. 2 0 < x < 3 means that 0 < x and x < 3, so we have p ∧ q.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Logically Dealing with Inequalities

Let p, q, and r symbolize x > 0, x < 3, and x = 3,

respectively. Write the following inequalities symbolically.

1 x ≤ 3 2 0 < x < 3 3 0 < x ≤ 3

Answers

1 x ≤ 3 means that x is less than or equal to 3, so we have q ∨ r. 2 0 < x < 3 means that 0 < x and x < 3, so we have p ∧ q. 3 0 < x ≤ 3 means that 0 < x and x ≤ 3, so we have p ∧ (q ∨ r).

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

∼p has the opposite truth value of p

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

∼p has the opposite truth value of p If p is true, ∼p is false, and if p is false, ∼p is true.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

∼p has the opposite truth value of p If p is true, ∼p is false, and if p is false, ∼p is true. Truth table for ∼p : p ∼p T F F T

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

p ∧ q is true when, and only when, both p and q are true.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

p ∧ q is true when, and only when, both p and q are true. If at least one of p or q is false, their conjunction p ∧ q is false as well.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

p ∧ q is true when, and only when, both p and q are true. If at least one of p or q is false, their conjunction p ∧ q is false as well. Truth table for p ∧ q : p q p ∧ q T T T T F F F T F F F F

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

p ∨ q is true whenever at least one of p and q are true.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

p ∨ q is true whenever at least one of p and q are true. p ∨ q is false when, and only when, p and q are both false

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

p ∨ q is true whenever at least one of p and q are true. p ∨ q is false when, and only when, p and q are both false Truth table for p ∨ q : p q p ∨ q T T T T F T F T T F F F

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

A statement form (or propositional form) is an expression made up of statement variables (such as p, q, and r) and logical connectives (such as ∼, ∧, and ∨) that becomes a statement when actual statements are substituted for the component statement variables.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

A statement form (or propositional form) is an expression made up of statement variables (such as p, q, and r) and logical connectives (such as ∼, ∧, and ∨) that becomes a statement when actual statements are substituted for the component statement variables. The truth table for a given statement form displays the truth values that correspond to all possible combinations of truth values for its component statement variables.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

A statement form (or propositional form) is an expression made up of statement variables (such as p, q, and r) and logical connectives (such as ∼, ∧, and ∨) that becomes a statement when actual statements are substituted for the component statement variables. The truth table for a given statement form displays the truth values that correspond to all possible combinations of truth values for its component statement variables. To compute the truth values for a statement form, follow rules similar to those used to evaluate algebraic expressions.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

A statement form (or propositional form) is an expression made up of statement variables (such as p, q, and r) and logical connectives (such as ∼, ∧, and ∨) that becomes a statement when actual statements are substituted for the component statement variables. The truth table for a given statement form displays the truth values that correspond to all possible combinations of truth values for its component statement variables. To compute the truth values for a statement form, follow rules similar to those used to evaluate algebraic expressions. For each combination of truth values for the statement variables, first evaluate the expressions within the innermost parentheses, and then work your way out until you have the truth values for the complete expression.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Note that p XOR q can be expressed as (p ∨ q) ∧ ∼(p ∧ q).

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Note that p XOR q can be expressed as (p ∨ q) ∧ ∼(p ∧ q). The truth table for “exclusive or” is thus p q p ∨ q p ∧ q ∼(p ∧ q) (p ∨ q) ∧ ∼(p ∧ q) T T T T F F T F T F T T F T T F T T F F F F T F

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Another Example

The truth table for (p ∧ q) ∨ ∼r is p q r p ∧ q ∼r (p ∧ q) ∨ ∼r T T T T F T T T F T T T T F T F F F T F F F T T F T T F F F F T F F T T F F T F F F F F F F T T

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Tautologies and Contradictions

A tautology is a statement form that’s always true regardless

f the truth values of the individual statements and is

sometimes denoted by t.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Tautologies and Contradictions

A tautology is a statement form that’s always true regardless

f the truth values of the individual statements and is

sometimes denoted by t. A contradiction is a statement form that’s always false regardless of the truth values of the individual statements and is sometimes denoted by c.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Tautologies and Contradictions

A tautology is a statement form that’s always true regardless

f the truth values of the individual statements and is

sometimes denoted by t. A contradiction is a statement form that’s always false regardless of the truth values of the individual statements and is sometimes denoted by c. e.g. p ∨ ∼p is a tautology whereas p ∧ ∼p is a contradiction: p ∼p p ∨ ∼p p ∧ ∼p T F T F F T T F

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Exercises

If a statement form has n variables, how many rows will its truth table have? Construct truth tables for the following statement forms:

1 ∼(p ∧ q) 2 ∼p ∧ ∼q 3 ∼p ∨ ∼q 4 p ∨ (∼q ∧ r)

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Two statement forms are called logically equivalent if, and

nly if, they have identical truth values for each possible

assignment of truth values to their statement variables.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Two statement forms are called logically equivalent if, and

nly if, they have identical truth values for each possible

assignment of truth values to their statement variables. Notation: If P and Q are logically equivalent, we write P ≡ Q.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Two statement forms are called logically equivalent if, and

nly if, they have identical truth values for each possible

assignment of truth values to their statement variables. Notation: If P and Q are logically equivalent, we write P ≡ Q. To test whether P and Q are logically equivalent:

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Two statement forms are called logically equivalent if, and

nly if, they have identical truth values for each possible

assignment of truth values to their statement variables. Notation: If P and Q are logically equivalent, we write P ≡ Q. To test whether P and Q are logically equivalent:

1 Construct a truth table with one column for P and another

column for Q.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Two statement forms are called logically equivalent if, and

nly if, they have identical truth values for each possible

assignment of truth values to their statement variables. Notation: If P and Q are logically equivalent, we write P ≡ Q. To test whether P and Q are logically equivalent:

1 Construct a truth table with one column for P and another

column for Q.

2 Check for whether these two columns are identical.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Example: double negation property: ∼(∼p) ≡ p p ∼p ∼(∼p) T F T F T F

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Example: De Morgan’s Laws p q ∼p ∼q p ∧ q ∼(p ∧ q) ∼p ∨ ∼q T T F F T F F T F F T F T T F T T F F T T F F T T F T T

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Example: De Morgan’s Laws p q ∼p ∼q p ∧ q ∼(p ∧ q) ∼p ∨ ∼q T T F F T F F T F F T F T T F T T F F T T F F T T F T T So ∼(p ∧ q) ≡ ∼p ∨ ∼q.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Example: De Morgan’s Laws p q ∼p ∼q p ∧ q ∼(p ∧ q) ∼p ∨ ∼q T T F F T F F T F F T F T T F T T F F T T F F T T F T T So ∼(p ∧ q) ≡ ∼p ∨ ∼q. Exercise: use truth tables to show that ∼(p ∨ q) ≡ ∼p ∧ ∼q.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

De Morgan’s Laws demystified

Simply stated, the negation of an “and” statement is logically equivalent to the “or” statement in which each component is negated.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

De Morgan’s Laws demystified

Simply stated, the negation of an “and” statement is logically equivalent to the “or” statement in which each component is negated. This should intuitively should make sense. In order for p ∧ q to be false, we would need p to be false or q to be false.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

De Morgan’s Laws demystified

Simply stated, the negation of an “and” statement is logically equivalent to the “or” statement in which each component is negated. This should intuitively should make sense. In order for p ∧ q to be false, we would need p to be false or q to be false. The negation of an “or” statement is logically equivalent to the “and” statement in which each component is negated.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

De Morgan’s Laws demystified

Simply stated, the negation of an “and” statement is logically equivalent to the “or” statement in which each component is negated. This should intuitively should make sense. In order for p ∧ q to be false, we would need p to be false or q to be false. The negation of an “or” statement is logically equivalent to the “and” statement in which each component is negated. Again, this should intuitively make sense. In order for p ∨ q to be false, both p and q need to be false.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Write negations of each of the following statements in simple English.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Write negations of each of the following statements in simple English.

1 John is 6 feet tall and he weighs at least 200 pounds.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Write negations of each of the following statements in simple English.

1 John is 6 feet tall and he weighs at least 200 pounds. 2 The bus was late or Tom’s watch was slow.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Write negations of each of the following statements in simple English.

1 John is 6 feet tall and he weighs at least 200 pounds. 2 The bus was late or Tom’s watch was slow.

Answers:

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Write negations of each of the following statements in simple English.

1 John is 6 feet tall and he weighs at least 200 pounds. 2 The bus was late or Tom’s watch was slow.

Answers:

1 John is not 6 feet tall or he weighs less than 200 pounds.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Write negations of each of the following statements in simple English.

1 John is 6 feet tall and he weighs at least 200 pounds. 2 The bus was late or Tom’s watch was slow.

Answers:

1 John is not 6 feet tall or he weighs less than 200 pounds. 2 The bus was not late and Tom’s watch was not slow.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Write negations of each of the following statements in simple English.

1 John is 6 feet tall and he weighs at least 200 pounds. 2 The bus was late or Tom’s watch was slow.

Answers:

1 John is not 6 feet tall or he weighs less than 200 pounds. 2 The bus was not late and Tom’s watch was not slow.

WARNING: “The bus was early and Tom’s watch was fast” is an incorrect negation! Be careful with what the opposite of common English words are.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications

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Propositional Logic Truth Tables Logical Equivalence

Common Logical Equivalences

Exercise: Verify these using truth tables.

Chapter 2: The Logic of Compound Statements (Part 1) Discrete Mathematics with Applications