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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. 1.2: Propositional Equivalences. Definition:

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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

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.


Logical equivalences

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 equivalences1

Logical Equivalences

One way to determine whether two compound propositions are equivalent is to use a truth table.

Symbol: PQ


Chapter 1 the foundations logic and proofs

Logical Equivalences

  • Prove the De Morgan’s Laws.


Chapter 1 the foundations logic and proofs

Logical Equivalences

  • HW: Prove the other one (De Morgan’s Laws).


Example show that p q and p q are logically equivalent hw example 4 of page 23

Logical Equivalences

Example:

Show that pq and ¬pqare logically equivalent.

HW: example 4 of page 23


T01 2 006 jpg

Logical Equivalences

t01_2_006.jpg




Chapter 1 the foundations logic and proofs

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”.


Chapter 1 the foundations logic and proofs

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.


Chapter 1 the foundations logic and proofs

Terms

  • Tautology

  • Contradiction

  • Contingency

  • Logical Equivalence

  • De Morgan’s Laws

  • Commutative Law

  • Associative Law

  • Distributive Law