Presentation on theme: "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."— Presentation transcript:
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 Double Negation ( p) p Domination p T T p F F Idempotent p p p
Important Equivalences Commutative p q q p Associative (p q) r p (q r) Distributive p (q r) (p q) (p r) De Morgans (p q) p q
Important Equivalences Absorption 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 q p (p q) (p r) p (q r) p q (p q) (q p)