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Chapter 1: The Foundations: Logic and Proofs

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:

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Chapter 1: The Foundations: Logic and Proofs

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  1. 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

  2. 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.

  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 pq is a tautology. Denote pq.

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

  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 pq and ¬pqare 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 ¬(pq) and p ¬q are logically equivalent. • Example 7: Show that ¬(p(¬p  q)) and ¬p  ¬q are logically equivalent by developing a serious of logical equivalences. • Example 8: Show that (p  q) (pq) is a tautology.

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

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