Chapter 1: The Foundations: Logic and Proofs

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Propositional Equivalences

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Presentation transcript:

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

1.2: 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.

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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 pq is a tautology. Denote pq.

Logical Equivalences One way to determine whether two compound propositions are equivalent is to use a truth table. Symbol: PQ

Logical Equivalences Prove the De Morgan’s Laws. p:\msoffice\My Projects\Rosen 6e 2007\Imagebank\JPEGs07-24-06\ch01\jpeg\t01_2_002.jpg

Logical Equivalences HW: Prove the other one (De Morgan’s Laws). p:\msoffice\My Projects\Rosen 6e 2007\Imagebank\JPEGs07-24-06\ch01\jpeg\t01_2_003.jpg HW: Prove the other one (De Morgan’s Laws).

Logical Equivalences Example: Show that pq and ¬pq are logically equivalent. HW: example 4 of page 23

Logical Equivalences t01_2_006.jpg p:\msoffice\My Projects\Rosen 6e 2007\Imagebank\JPEGs07-24-06\ch01\jpeg\t01_2_006.jpg

Logical Equivalences p:\msoffice\My Projects\Rosen 6e 2007\Imagebank\JPEGs07-24-06\ch01\jpeg\t01_2_007.jpg

Logical Equivalences p:\msoffice\My Projects\Rosen 6e 2007\Imagebank\JPEGs07-24-06\ch01\jpeg\t01_2_008.jpg

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”.

Logical Equivalences Example 6: Show that ¬(pq) 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) ( pq) is a tautology.

Terms Tautology Contradiction Contingency Logical Equivalence De Morgan’s Laws Commutative Law Associative Law Distributive Law