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Chapter 1: The Foundations: Logic and Proofs 1.1 Propositional Logic 1.2 Propositional Equivalences 1.3 Predicates and Quantifiers 1.4 Nested Quantifiers 1.5 Rules of Inference 1.6 Introduction to Proofs 1.7 Proof Methods and Strategy 1 1.2: Propositional Equivalences Definition: Tautology: A compound proposition that is always true. Contradiction: A compound proposition that is always false. Contingency: A compound proposition that is neither a tautology nor a contradiction. 2 3 Logical Equivalences • Compound propositions that have the same truth values in all possible cases are called logically equivalent. • Definition: The compound propositions p and q are called logically equivalent if pq is a tautology. Denote pq. 4 Logical Equivalences • • One way to determine whether two compound propositions are equivalent is to use a truth table. Symbol: PQ 5 Logical Equivalences • Prove the De Morgan’s Laws. 6 Logical Equivalences • HW: Prove the other one (De Morgan’s Laws). 7 Logical Equivalences • Example: Show that pq and ¬pq are logically equivalent. • HW: example 4 of page 23 8 Logical Equivalences t01_2_006.jpg 9 Logical Equivalences 10 Logical Equivalences 11 Logical Equivalences Example 5: Use De Morgan’s laws to express the negations of “Miguel has a cellphone and he has a laptop computer”. Example 5: Use De Morgan’s laws to express the negations of “Heather will go to the concert or Steve will go to the concert”. 12 Logical Equivalences • Example 6: Show that ¬(pq) and p ¬q are logically equivalent. • Example 7: Show that ¬(p(¬p q)) and ¬p ¬q are logically equivalent by developing a series of logical equivalences. • Example 8: Show that (p q) ( pq) is a tautology. 13 Terms • • • • • • • • Tautology Contradiction Contingency Logical Equivalence De Morgan’s Laws Commutative Law Associative Law Distributive Law 14