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# Intro to Proofs

### Logic, Laws of Inference, etc.

TermDefinition
Conjunction statements combined with "and", p^q, true when both p and q are TRUE.
Disjunction statements combined with "or", pVq, false when both p and q are FALSE.
Conditional if-then statements, p->q, false when p is true but q is FALSE.
Biconditional if and only if, p<->q, true when both p and q have same TRUTH VALUE.
Negation not, ~p (or ~q), opposites
Law of Detachment ex: If a person is a tennis player , then he/she is an athlete (p->q). Serena Williams is a tennis player (p). ∴ Serena Williams is an athlete (∴q).
Law of Syllogism If you were born in Miami Beach, then you were born in Florida (p->q). If you were born in Florida, then you a native Floridian (q->r). ∴ If you were born in Miami Beach, then you are a native Floridian (∴p->r).
Law of Simplification p: You like ice cream. q: You like cheeseburgers. You like ice cream and you like cheeseburgers (p^q). ∴ You like ice cream (∴p). You like ice cream and you like cheeseburgers (p^q). ∴ You like cheeseburgers (∴q).
Law of Contrapositive Inference p: You are a voter. q: You are of legal age. If you are a voter, then you are of legal age (p->q). You are not of legal age (~q). ∴ You are not a voter (∴~p).
Law of Disjunctive Inference p: You like burgers. q: You like pasta. You like burgers or you like pasta (pVq). You don't like cheeseburgers (~p). ∴ You like pasta (∴q). You like burgers s or you like pasta (pVq). You don't like pasta (~q). ∴ You like burgers (∴p).
DeMorgan's Law p: He has a car. q: He has a bus. "He either doesn't have a car, or doesn't have a bus, idk which, maybe he has neither" [ ~(p^q)<->(~pV~q) ] "He doesn't have a car, & he doesn't have a bus" [ ~(pVq)<->(~p^~q) ]
Created by: ka0624