Propositional Equivalences Section 1.2

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

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

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

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.

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

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

Important Equivalences Absorption p  (p  q)  p p  (p  q)  p Negation p  p  T p  p  F

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

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)

Example Show that (p  q)  (p  q) is a tautology.

Next Lecture 1.3 Predicates and Quantifiers