Overview
Boolean Algebra
Implications and Negations
Negation
Bernd Schröder
Bernd Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Uses of Negation
Bernd Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Boolean Algebra, Part II
Bernd Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Boolean Algebra, Part II
Theorem. DeMorgan’s Laws. Let p, q be primitive
propositions.
Bernd Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Verbal Proof of ¬(p ∧ q) = (¬p) ∨ (¬q)
Bernd Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Algebraic Proof of ¬(p ∧ q) = (¬p) ∨ (¬q)
Bernd Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Algebraic Proof of ¬(p ∧ q) = (¬p) ∨ (¬q)
¬(p ∧ q)
Bernd Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Algebraic Proof of ¬(p ∧ q) = (¬p) ∨ (¬q)
¬(p ∧ q) = (¬p ∧ ¬q)
Bernd Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Algebraic Proof of ¬(p ∧ q) = (¬p) ∨ (¬q)
¬(p ∧ q) = (¬p ∧ ¬q) ∨ (¬p ∧ q)
Bernd Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Algebraic Proof of ¬(p ∧ q) = (¬p) ∨ (¬q)
¬(p ∧ q) = (¬p ∧ ¬q) ∨ (¬p ∧ q) ∨ (p ∧ ¬q)
Bernd Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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)
| {z }
=TRUE
Bernd Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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)
| {z }
=TRUE
= ¬p ∨ (p ∧ ¬q)
Bernd Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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)
| {z }
=TRUE
= ¬p ∨ (p ∧ ¬q)
= (¬p ∨ p) ∧ (¬p ∨ ¬q)
Bernd Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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)
| {z }
=TRUE
= ¬p ∨ (p ∧ ¬q)
= (¬p ∨ p) ∧(¬p ∨ ¬q)
| {z }
=TRUE
Bernd Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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)
| {z }
=TRUE
= ¬p ∨ (p ∧ ¬q)
= (¬p ∨ p) ∧(¬p ∨ ¬q)
| {z }
=TRUE
= ¬p ∨ ¬q
Bernd Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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)
| {z }
=TRUE
= ¬p ∨ (p ∧ ¬q)
= (¬p ∨ p) ∧(¬p ∨ ¬q)
| {z }
=TRUE
= ¬p ∨ ¬q
Bernd Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
Bernd Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
Bernd Schröder
Negation
q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
Bernd Schröder
Negation
q
p∧q
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Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
Bernd Schröder
Negation
q
p∧q
¬(p ∧ q)
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
Bernd Schröder
Negation
q
p∧q
¬(p ∧ q)
¬p
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
Bernd Schröder
Negation
q
p∧q
¬(p ∧ q)
¬p
¬q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
Bernd Schröder
Negation
q
p∧q
¬(p ∧ q)
¬p
¬q
¬p ∨ ¬q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
Bernd Schröder
Negation
q
FALSE
p∧q
¬(p ∧ q)
¬p
¬q
¬p ∨ ¬q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
FALSE
TRUE
Bernd Schröder
Negation
p∧q
¬(p ∧ q)
¬p
¬q
¬p ∨ ¬q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
FALSE
TRUE
TRUE
FALSE
Bernd Schröder
Negation
p∧q
¬(p ∧ q)
¬p
¬q
¬p ∨ ¬q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
FALSE
TRUE
TRUE
FALSE
TRUE
TRUE
Bernd Schröder
Negation
p∧q
¬(p ∧ q)
¬p
¬q
¬p ∨ ¬q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
FALSE
TRUE
TRUE
FALSE
TRUE
TRUE
Bernd Schröder
Negation
p∧q
FALSE
¬(p ∧ q)
¬p
¬q
¬p ∨ ¬q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
p∧q
FALSE
FALSE
TRUE
FALSE
TRUE
FALSE
TRUE
TRUE
Bernd Schröder
Negation
¬(p ∧ q)
¬p
¬q
¬p ∨ ¬q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
p∧q
FALSE
FALSE
TRUE
FALSE
TRUE
FALSE
FALSE
TRUE
TRUE
Bernd Schröder
Negation
¬(p ∧ q)
¬p
¬q
¬p ∨ ¬q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
p∧q
FALSE
FALSE
TRUE
FALSE
TRUE
FALSE
FALSE
TRUE
TRUE
TRUE
Bernd Schröder
Negation
¬(p ∧ q)
¬p
¬q
¬p ∨ ¬q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
p∧q
FALSE
FALSE
TRUE
FALSE
TRUE
FALSE
FALSE
TRUE
TRUE
TRUE
Bernd Schröder
Negation
¬(p ∧ q)
TRUE
¬p
¬q
¬p ∨ ¬q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
p∧q
FALSE
¬(p ∧ q)
TRUE
FALSE
TRUE
FALSE
TRUE
TRUE
FALSE
FALSE
TRUE
TRUE
TRUE
Bernd Schröder
Negation
¬p
¬q
¬p ∨ ¬q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
p∧q
FALSE
¬(p ∧ q)
TRUE
FALSE
TRUE
FALSE
TRUE
TRUE
FALSE
FALSE
TRUE
TRUE
TRUE
TRUE
Bernd Schröder
Negation
¬p
¬q
¬p ∨ ¬q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
p∧q
FALSE
¬(p ∧ q)
TRUE
FALSE
TRUE
FALSE
TRUE
TRUE
FALSE
FALSE
TRUE
TRUE
TRUE
TRUE
FALSE
Bernd Schröder
Negation
¬p
¬q
¬p ∨ ¬q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
p∧q
FALSE
¬(p ∧ q)
TRUE
FALSE
TRUE
FALSE
TRUE
TRUE
FALSE
FALSE
TRUE
TRUE
TRUE
TRUE
FALSE
Bernd Schröder
Negation
¬p
TRUE
¬q
¬p ∨ ¬q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
p∧q
FALSE
¬(p ∧ q)
TRUE
¬p
TRUE
FALSE
TRUE
FALSE
TRUE
TRUE
TRUE
FALSE
FALSE
TRUE
TRUE
TRUE
TRUE
FALSE
Bernd Schröder
Negation
¬q
¬p ∨ ¬q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
p∧q
FALSE
¬(p ∧ q)
TRUE
¬p
TRUE
FALSE
TRUE
FALSE
TRUE
TRUE
TRUE
FALSE
FALSE
TRUE
FALSE
TRUE
TRUE
TRUE
FALSE
Bernd Schröder
Negation
¬q
¬p ∨ ¬q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
p∧q
FALSE
¬(p ∧ q)
TRUE
¬p
TRUE
FALSE
TRUE
FALSE
TRUE
TRUE
TRUE
FALSE
FALSE
TRUE
FALSE
TRUE
TRUE
TRUE
FALSE
FALSE
Bernd Schröder
Negation
¬q
¬p ∨ ¬q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
p∧q
FALSE
¬(p ∧ q)
TRUE
¬p
TRUE
FALSE
TRUE
FALSE
TRUE
TRUE
TRUE
FALSE
FALSE
TRUE
FALSE
TRUE
TRUE
TRUE
FALSE
FALSE
Bernd Schröder
Negation
¬q
TRUE
¬p ∨ ¬q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
p∧q
FALSE
¬(p ∧ q)
TRUE
¬p
TRUE
¬q
TRUE
FALSE
TRUE
FALSE
TRUE
TRUE
FALSE
TRUE
FALSE
FALSE
TRUE
FALSE
TRUE
TRUE
TRUE
FALSE
FALSE
Bernd Schröder
Negation
¬p ∨ ¬q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
p∧q
FALSE
¬(p ∧ q)
TRUE
¬p
TRUE
¬q
TRUE
FALSE
TRUE
FALSE
TRUE
TRUE
FALSE
TRUE
FALSE
FALSE
TRUE
FALSE
TRUE
TRUE
TRUE
TRUE
FALSE
FALSE
Bernd Schröder
Negation
¬p ∨ ¬q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
p∧q
FALSE
¬(p ∧ q)
TRUE
¬p
TRUE
¬q
TRUE
FALSE
TRUE
FALSE
TRUE
TRUE
FALSE
TRUE
FALSE
FALSE
TRUE
FALSE
TRUE
TRUE
TRUE
TRUE
FALSE
FALSE
FALSE
Bernd Schröder
Negation
¬p ∨ ¬q
logo1
Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
p∧q
FALSE
¬(p ∧ q)
TRUE
¬p
TRUE
¬q
TRUE
FALSE
TRUE
FALSE
TRUE
TRUE
FALSE
TRUE
FALSE
FALSE
TRUE
FALSE
TRUE
TRUE
TRUE
TRUE
FALSE
FALSE
FALSE
Bernd Schröder
Negation
¬p ∨ ¬q
TRUE
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Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
p∧q
FALSE
¬(p ∧ q)
TRUE
¬p
TRUE
¬q
TRUE
¬p ∨ ¬q
TRUE
FALSE
TRUE
FALSE
TRUE
TRUE
FALSE
TRUE
TRUE
FALSE
FALSE
TRUE
FALSE
TRUE
TRUE
TRUE
TRUE
FALSE
FALSE
FALSE
Bernd Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
p∧q
FALSE
¬(p ∧ q)
TRUE
¬p
TRUE
¬q
TRUE
¬p ∨ ¬q
TRUE
FALSE
TRUE
FALSE
TRUE
TRUE
FALSE
TRUE
TRUE
FALSE
FALSE
TRUE
FALSE
TRUE
TRUE
TRUE
TRUE
TRUE
FALSE
FALSE
FALSE
Bernd Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
p∧q
FALSE
¬(p ∧ q)
TRUE
¬p
TRUE
¬q
TRUE
¬p ∨ ¬q
TRUE
FALSE
TRUE
FALSE
TRUE
TRUE
FALSE
TRUE
TRUE
FALSE
FALSE
TRUE
FALSE
TRUE
TRUE
TRUE
TRUE
TRUE
FALSE
FALSE
FALSE
FALSE
Bernd Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Proof of ¬(p ∧ q) = (¬p) ∨ (¬q) With a Truth
Table
p
FALSE
q
FALSE
p∧q
FALSE
¬(p ∧ q)
TRUE
¬p
TRUE
¬q
TRUE
¬p ∨ ¬q
TRUE
FALSE
TRUE
FALSE
TRUE
TRUE
FALSE
TRUE
TRUE
FALSE
FALSE
TRUE
FALSE
TRUE
TRUE
TRUE
TRUE
TRUE
FALSE
FALSE
FALSE
FALSE
Bernd Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Double Negation
Bernd Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
The Contrapositive
Bernd Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
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 6∈ N.
Bernd Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
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 6∈ N.
Contrapositive:
Bernd Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
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 6∈ N.
Contrapositive:
If c ∈ N, then c3 6= a3 + b3 .
Bernd Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
Overview
Boolean Algebra
Implications and Negations
Negating Implications
Bernd Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
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 Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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 6∈ N.
Bernd Schröder
Negation
logo1
Louisiana Tech University, College of Engineering and Science
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 6∈ N.
Negation.
Bernd Schröder
Negation
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Louisiana Tech University, College of Engineering and Science
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 6∈ N.
Negation.
c2 = a2 + b2 and c ∈ N.
Bernd Schröder
Negation
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Louisiana Tech University, College of Engineering and Science