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Negation Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech - - PowerPoint PPT Presentation

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Overview Boolean Algebra Implications and Negations Negation Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science Negation Overview Boolean Algebra Implications and Negations Uses of

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logo1 Overview Boolean Algebra Implications and Negations

Negation

Bernd Schr¨

  • der

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Uses of Negation

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Uses of Negation

  • 1. A proof of p ⇒ q by contradiction starts with the negation

¬q of the conclusion q and leads this statement to a contradiction.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Uses of Negation

  • 1. A proof of p ⇒ q by contradiction starts with the negation

¬q of the conclusion q and leads this statement to a contradiction.

  • 2. Similarly, a proof by contraposition requires correct

negations.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Uses of Negation

  • 1. A proof of p ⇒ q by contradiction starts with the negation

¬q of the conclusion q and leads this statement to a contradiction.

  • 2. Similarly, a proof by contraposition requires correct

negations.

  • 3. So we must carefully study negations of logical

connectives.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Boolean Algebra, Part II

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Boolean Algebra, Part II

  • Theorem. DeMorgan’s Laws. Let p,q be primitive

propositions.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Boolean Algebra, Part II

  • Theorem. DeMorgan’s Laws. Let p,q be primitive

propositions.

  • 1. The negation of the statement p∧q is

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Boolean Algebra, Part II

  • Theorem. DeMorgan’s Laws. Let p,q be primitive

propositions.

  • 1. The negation of the statement p∧q is

¬(p∧q)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 10

logo1 Overview Boolean Algebra Implications and Negations

Boolean Algebra, Part II

  • Theorem. DeMorgan’s Laws. Let p,q be primitive

propositions.

  • 1. The negation of the statement p∧q is

¬(p∧q) = (¬p)∨(¬q).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 11

logo1 Overview Boolean Algebra Implications and Negations

Boolean Algebra, Part II

  • Theorem. DeMorgan’s Laws. Let p,q be primitive

propositions.

  • 1. The negation of the statement p∧q is

¬(p∧q) = (¬p)∨(¬q).

  • 2. The negation of the statement p∨q is

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 12

logo1 Overview Boolean Algebra Implications and Negations

Boolean Algebra, Part II

  • Theorem. DeMorgan’s Laws. Let p,q be primitive

propositions.

  • 1. The negation of the statement p∧q is

¬(p∧q) = (¬p)∨(¬q).

  • 2. The negation of the statement p∨q is

¬(p∨q)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 13

logo1 Overview Boolean Algebra Implications and Negations

Boolean Algebra, Part II

  • Theorem. DeMorgan’s Laws. Let p,q be primitive

propositions.

  • 1. The negation of the statement p∧q is

¬(p∧q) = (¬p)∨(¬q).

  • 2. The negation of the statement p∨q is

¬(p∨q) = (¬p)∧(¬q).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Verbal Proof of ¬(p∧q) = (¬p)∨(¬q)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Verbal Proof of ¬(p∧q) = (¬p)∨(¬q)

¬(p∧q) is false iff both p and q are true.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Verbal Proof of ¬(p∧q) = (¬p)∨(¬q)

¬(p∧q) is false iff both p and q are true. (¬p)∨(¬q) is false iff both p and q are true.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Verbal Proof of ¬(p∧q) = (¬p)∨(¬q)

¬(p∧q) is false iff both p and q are true. (¬p)∨(¬q) is false iff both p and q are true. Hence ¬(p∧q) is false iff (¬p)∨(¬q) is false.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Verbal Proof of ¬(p∧q) = (¬p)∨(¬q)

¬(p∧q) is false iff both p and q are true. (¬p)∨(¬q) is false iff both p and q are true. Hence ¬(p∧q) is false iff (¬p)∨(¬q) is

  • false. Consequently, ¬(p∧q) is true iff (¬p)∨(¬q) is true.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Verbal Proof of ¬(p∧q) = (¬p)∨(¬q)

¬(p∧q) is false iff both p and q are true. (¬p)∨(¬q) is false iff both p and q are true. Hence ¬(p∧q) is false iff (¬p)∨(¬q) is

  • false. Consequently, ¬(p∧q) is true iff (¬p)∨(¬q) is true. So

the two are equal.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Verbal Proof of ¬(p∧q) = (¬p)∨(¬q)

¬(p∧q) is false iff both p and q are true. (¬p)∨(¬q) is false iff both p and q are true. Hence ¬(p∧q) is false iff (¬p)∨(¬q) is

  • false. Consequently, ¬(p∧q) is true iff (¬p)∨(¬q) is true. So

the two are equal.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Verbal Proof of ¬(p∧q) = (¬p)∨(¬q)

¬(p∧q) is false iff both p and q are true. (¬p)∨(¬q) is false iff both p and q are true. Hence ¬(p∧q) is false iff (¬p)∨(¬q) is

  • false. Consequently, ¬(p∧q) is true iff (¬p)∨(¬q) is true. So

the two are equal. Almost feels like a truth table, but it highlights an important feature of proofs:

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Verbal Proof of ¬(p∧q) = (¬p)∨(¬q)

¬(p∧q) is false iff both p and q are true. (¬p)∨(¬q) is false iff both p and q are true. Hence ¬(p∧q) is false iff (¬p)∨(¬q) is

  • false. Consequently, ¬(p∧q) is true iff (¬p)∨(¬q) is true. So

the two are equal. Almost feels like a truth table, but it highlights an important feature of proofs: Once you find a “weak spot”, you can drive the argument to its conclusion.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Algebraic Proof of ¬(p∧q) = (¬p)∨(¬q)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Algebraic Proof of ¬(p∧q) = (¬p)∨(¬q)

¬(p∧q)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Algebraic Proof of ¬(p∧q) = (¬p)∨(¬q)

¬(p∧q) = (¬p∧¬q)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Algebraic Proof of ¬(p∧q) = (¬p)∨(¬q)

¬(p∧q) = (¬p∧¬q)∨(¬p∧q)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Algebraic Proof of ¬(p∧q) = (¬p)∨(¬q)

¬(p∧q) = (¬p∧¬q)∨(¬p∧q)∨(p∧¬q)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Algebraic Proof of ¬(p∧q) = (¬p)∨(¬q)

¬(p∧q) = (¬p∧¬q)∨(¬p∧q)∨(p∧¬q) =

  • (¬p∧¬q)∨(¬p∧q)
  • ∨(p∧¬q)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Algebraic Proof of ¬(p∧q) = (¬p)∨(¬q)

¬(p∧q) = (¬p∧¬q)∨(¬p∧q)∨(p∧¬q) =

  • (¬p∧¬q)∨(¬p∧q)
  • ∨(p∧¬q)

=

  • ¬p∧(¬q∨q)
  • ∨(p∧¬q)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Algebraic Proof of ¬(p∧q) = (¬p)∨(¬q)

¬(p∧q) = (¬p∧¬q)∨(¬p∧q)∨(p∧¬q) =

  • (¬p∧¬q)∨(¬p∧q)
  • ∨(p∧¬q)

=

  • ¬p∧(¬q∨q)

=TRUE

  • ∨(p∧¬q)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Algebraic Proof of ¬(p∧q) = (¬p)∨(¬q)

¬(p∧q) = (¬p∧¬q)∨(¬p∧q)∨(p∧¬q) =

  • (¬p∧¬q)∨(¬p∧q)
  • ∨(p∧¬q)

=

  • ¬p∧(¬q∨q)

=TRUE

  • ∨(p∧¬q)

= ¬p∨(p∧¬q)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Algebraic Proof of ¬(p∧q) = (¬p)∨(¬q)

¬(p∧q) = (¬p∧¬q)∨(¬p∧q)∨(p∧¬q) =

  • (¬p∧¬q)∨(¬p∧q)
  • ∨(p∧¬q)

=

  • ¬p∧(¬q∨q)

=TRUE

  • ∨(p∧¬q)

= ¬p∨(p∧¬q) = (¬p∨p)∧(¬p∨¬q)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Algebraic Proof of ¬(p∧q) = (¬p)∨(¬q)

¬(p∧q) = (¬p∧¬q)∨(¬p∧q)∨(p∧¬q) =

  • (¬p∧¬q)∨(¬p∧q)
  • ∨(p∧¬q)

=

  • ¬p∧(¬q∨q)

=TRUE

  • ∨(p∧¬q)

= ¬p∨(p∧¬q) = (¬p∨p)

=TRUE

∧(¬p∨¬q)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Algebraic Proof of ¬(p∧q) = (¬p)∨(¬q)

¬(p∧q) = (¬p∧¬q)∨(¬p∧q)∨(p∧¬q) =

  • (¬p∧¬q)∨(¬p∧q)
  • ∨(p∧¬q)

=

  • ¬p∧(¬q∨q)

=TRUE

  • ∨(p∧¬q)

= ¬p∨(p∧¬q) = (¬p∨p)

=TRUE

∧(¬p∨¬q) = ¬p∨¬q

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 35

logo1 Overview Boolean Algebra Implications and Negations

Algebraic Proof of ¬(p∧q) = (¬p)∨(¬q)

¬(p∧q) = (¬p∧¬q)∨(¬p∧q)∨(p∧¬q) =

  • (¬p∧¬q)∨(¬p∧q)
  • ∨(p∧¬q)

=

  • ¬p∧(¬q∨q)

=TRUE

  • ∨(p∧¬q)

= ¬p∨(p∧¬q) = (¬p∨p)

=TRUE

∧(¬p∨¬q) = ¬p∨¬q

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 36

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 37

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

p q

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 38

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

p q p∧q

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 39

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

p q p∧q ¬(p∧q)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 40

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

p q p∧q ¬(p∧q) ¬p

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 41

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

p q p∧q ¬(p∧q) ¬p ¬q

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 42

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

p q p∧q ¬(p∧q) ¬p ¬q ¬p∨¬q

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

p q p∧q ¬(p∧q) ¬p ¬q ¬p∨¬q FALSE FALSE

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 44

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 45

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 46

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 47

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 48

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

p q p∧q ¬(p∧q) ¬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 Negation

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SLIDE 49

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

p q p∧q ¬(p∧q) ¬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 Negation

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SLIDE 50

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

p q p∧q ¬(p∧q) ¬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 Negation

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SLIDE 51

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 52

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 53

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 54

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 55

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 56

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 57

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 58

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 59

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 60

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 61

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 62

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 63

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 64

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 65

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 67

logo1 Overview Boolean Algebra Implications and Negations

Proof of ¬(p∧q) = (¬p)∨(¬q) With a Truth Table

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

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 68

logo1 Overview Boolean Algebra Implications and Negations

Double Negation

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 69

logo1 Overview Boolean Algebra Implications and Negations

Double Negation

  • Proposition. Law of double negation. Let p be a primitive
  • proposition. The negation of the statement ¬p is ¬(¬p) = p.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 70

logo1 Overview Boolean Algebra Implications and Negations

Double Negation

  • Proposition. Law of double negation. Let p be a primitive
  • proposition. The negation of the statement ¬p is ¬(¬p) = p.

Easily verified with truth tables.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 71

logo1 Overview Boolean Algebra Implications and Negations

Double Negation

  • Proposition. Law of double negation. Let p be a primitive
  • proposition. The negation of the statement ¬p is ¬(¬p) = p.

Easily verified with truth tables. “I don’t got nothing to say” would mean that I have something to say.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 72

logo1 Overview Boolean Algebra Implications and Negations

Double Negation

  • Proposition. Law of double negation. Let p be a primitive
  • proposition. The negation of the statement ¬p is ¬(¬p) = p.

Easily verified with truth tables. “I don’t got nothing to say” would mean that I have something to say. “I don’t got no nothing to say” would mean that I have nothing to say.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 73

logo1 Overview Boolean Algebra Implications and Negations

The Contrapositive

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 74

logo1 Overview Boolean Algebra Implications and Negations

The Contrapositive

  • Definition. Let p and q be primitive propositions. Then the

contrapositive of p ⇒ q is ¬q ⇒ ¬p.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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logo1 Overview Boolean Algebra Implications and Negations

The Contrapositive

  • Definition. Let p and q be primitive propositions. Then the

contrapositive of p ⇒ q is ¬q ⇒ ¬p. The contrapositive is equivalent to the original statement.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 76

logo1 Overview Boolean Algebra Implications and Negations

The Contrapositive

  • Definition. Let p and q be primitive propositions. Then the

contrapositive of p ⇒ q is ¬q ⇒ ¬p. The contrapositive is equivalent to the original statement.

  • Example. Let a,b ∈ N.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 77

logo1 Overview Boolean Algebra Implications and Negations

The Contrapositive

  • Definition. Let p and q be primitive propositions. Then the

contrapositive of p ⇒ q is ¬q ⇒ ¬p. The contrapositive is equivalent to the original statement.

  • Example. Let a,b ∈ N.

If c3 = a3 +b3, then c ∈ N.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 78

logo1 Overview Boolean Algebra Implications and Negations

The Contrapositive

  • Definition. Let p and q be primitive propositions. Then the

contrapositive of p ⇒ q is ¬q ⇒ ¬p. The contrapositive is equivalent to the original statement.

  • Example. Let a,b ∈ N.

If c3 = a3 +b3, then c ∈ N. Contrapositive:

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 79

logo1 Overview Boolean Algebra Implications and Negations

The Contrapositive

  • Definition. Let p and q be primitive propositions. Then the

contrapositive of p ⇒ q is ¬q ⇒ ¬p. The contrapositive is equivalent to the original statement.

  • Example. Let a,b ∈ N.

If c3 = a3 +b3, then c ∈ N. Contrapositive: If c ∈ N, then c3 = a3 +b3.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 80

logo1 Overview Boolean Algebra Implications and Negations

Negating Implications

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 81

logo1 Overview Boolean Algebra Implications and Negations

Negating Implications

  • Proposition. Let p and q be primitive propositions. The

negation of the statement p ⇒ q is ¬(p ⇒ q) = p∧¬q.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 82

logo1 Overview Boolean Algebra Implications and Negations

Negating Implications

  • Proposition. Let p and q be primitive propositions. The

negation of the statement p ⇒ q is ¬(p ⇒ q) = p∧¬q. This result will be used any time we disprove an implication with a counterexample.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 83

logo1 Overview Boolean Algebra Implications and Negations

Negating Implications

  • Proposition. Let p and q be primitive propositions. The

negation of the statement p ⇒ q is ¬(p ⇒ q) = p∧¬q. This result will be used any time we disprove an implication with a counterexample.

  • Example. Let a,b ∈ N.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 84

logo1 Overview Boolean Algebra Implications and Negations

Negating Implications

  • Proposition. Let p and q be primitive propositions. The

negation of the statement p ⇒ q is ¬(p ⇒ q) = p∧¬q. This result will be used any time we disprove an implication with a counterexample.

  • Example. Let a,b ∈ N.

If c2 = a2 +b2, then c ∈ N.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 85

logo1 Overview Boolean Algebra Implications and Negations

Negating Implications

  • Proposition. Let p and q be primitive propositions. The

negation of the statement p ⇒ q is ¬(p ⇒ q) = p∧¬q. This result will be used any time we disprove an implication with a counterexample.

  • Example. Let a,b ∈ N.

If c2 = a2 +b2, then c ∈ N. Negation.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation

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SLIDE 86

logo1 Overview Boolean Algebra Implications and Negations

Negating Implications

  • Proposition. Let p and q be primitive propositions. The

negation of the statement p ⇒ q is ¬(p ⇒ q) = p∧¬q. This result will be used any time we disprove an implication with a counterexample.

  • Example. Let a,b ∈ N.

If c2 = a2 +b2, then c ∈ N. Negation. c2 = a2 +b2 and c ∈ N.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Negation