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.

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4. 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.

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

6. Logical Equivalences Prove the De Morgan’s Laws.
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7. Logical Equivalences HW: Prove the other one (De Morgan’s Laws).
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HW: Prove the other one (De Morgan’s Laws).

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

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10. Logical Equivalences
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11. Logical Equivalences
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12. 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”.

13. 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.

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