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**Propositional Equivalences**

Section 1.2

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Example You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old.

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**Basic Terminology A tautology is a proposition which is always true.**

p p A contradiction is a proposition that is always false. p p A contingency is a proposition that is neither a tautology nor a contradiction. p q r

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Logical Equivalences Two propositions p and q are logically equivalent if they have the same truth values in all possible cases. Two propositions p and q are logically equivalent if p q is a tautology. Notation: p q or p q

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**Determining Logical Equivalence**

Use a truth table. Show that (p q) and p q are logically equivalent. Not a very efficient method, WHY? Solution: Develop a series of equivalences.

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**Important Equivalences**

Identity p T p p F p Domination p T T p F F Idempotent p p p p p p Double Negation ( p) p

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**Important Equivalences**

Commutative p q q p p q q p Associative (p q) r p (q r) (p q) r p (q r) Distributive p (q r) (p q) (p r) p (q r) (p q) (p r) De Morgan’s (p q) p q (p q) p q

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**Important Equivalences**

Absorption p (p q) p p (p q) p Negation p p T p p F

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Example Show that (p (p q)) and p q are logically equivalent.

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**Important Equivalences Involving Implications**

p → q p q p → q q → p (p → q) (p → r) p → (q r) (p → q) (p → r) p → (q r) p↔ q (p → q) (q → p)

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Example Show that (p q) (p q) is a tautology.

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Next Lecture 1.3 Predicates and Quantifiers

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Chapter 1: The Foundations: Logic and Proofs

Chapter 1: The Foundations: Logic and Proofs

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