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

Identityp T pp F p

Double Negation( p) p

Dominationp T Tp F F

Idempotentp p pp p p

Important Equivalences

Commutativep q q pp q q p

Associative(p q) r p (q r)(p q) r p (q r)

Distributivep (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

Absorptionp (p q) pp (p q) p

Negationp p Tp p F

Example

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

Important Equivalences Involving Implications

p → q p qp → 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