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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 Negation logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

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

Overview Boolean Algebra Implications and Negations Boolean Algebra, Part II Theorem. DeMorgan’s Laws . Let p , q be primitive propositions. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

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 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

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 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

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 ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

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 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

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 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

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 ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

Overview Boolean Algebra Implications and Negations Verbal Proof of ¬ ( p ∧ q ) = ( ¬ p ) ∨ ( ¬ q ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

Overview Boolean Algebra Implications and Negations Verbal Proof of ¬ ( p ∧ q ) = ( ¬ p ) ∨ ( ¬ q ) ¬ ( p ∧ q ) is false iff both p and q are true. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

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: logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

Overview Boolean Algebra Implications and Negations Algebraic Proof of ¬ ( p ∧ q ) = ( ¬ p ) ∨ ( ¬ q ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

Overview Boolean Algebra Implications and Negations Algebraic Proof of ¬ ( p ∧ q ) = ( ¬ p ) ∨ ( ¬ q ) ¬ ( p ∧ q ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

Overview Boolean Algebra Implications and Negations Algebraic Proof of ¬ ( p ∧ q ) = ( ¬ p ) ∨ ( ¬ q ) ¬ ( p ∧ q ) = ( ¬ p ∧¬ q ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

Overview Boolean Algebra Implications and Negations Algebraic Proof of ¬ ( p ∧ q ) = ( ¬ p ) ∨ ( ¬ q ) ¬ ( p ∧ q ) = ( ¬ p ∧¬ q ) ∨ ( ¬ p ∧ q ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

Overview Boolean Algebra Implications and Negations Algebraic Proof of ¬ ( p ∧ q ) = ( ¬ p ) ∨ ( ¬ q ) ¬ ( p ∧ q ) = ( ¬ p ∧¬ q ) ∨ ( ¬ p ∧ q ) ∨ ( p ∧¬ q ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation

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 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Negation