CSE 2353 August 28 th 2002 Logic Propositions

CSE 2353 – August 28 th 2002 Logic

Propositions • Def: A statement which is True or False • Which are Propositions – This rose is white – Keep off the grass – 7 is even – I am 6 years old – A square has four sides – Did you do your homework

Operations • And (conjunction) ^ p F F T T q F T p^q

Operations • • And (conjunction) ^ Or (inclusive disjunction) v Xor (exclusive disjunction) v Not (negation) ~ If … then (implication) Iff (biconditional) Nand | Nor • Construct sentences with words

Operations • Other Operations p F F T T q F T p^q p v q F F F T p q

Precedence: () ~ ^ v -> (Tv. F)^T = ? Tv(F^T) = ? Tv. F^T=?

Operations p F F T T q F T F T r F F T T p -> (q^r)

Operations p F F T T q F T F T r F F T T (~p v q) ~r

Tautologies and Contradictions • Tautology: A statement which is True • Contradiction: • p ^ ~p = • p v ~p = • (p^q) v ~(p^q) = • p -> p v q =

Practice • Show that (p q ) implies p

Practice • • • Show that p v q is equivalent to ~(~p ^ ~q) Show that p^p p Show that p q q p Show that (p ^ q) ^ r p ^ (q ^ r) Show that (p q) (~p v q)

Summary • • • Propositions Operations on Propositions Truth Tables Precedence Tautologies and Contradictions

Duality • Swap t, f • Swap ^, v • Example (p ^ q) v ~p • Dual (p v q) ^ ~p • If expressions are equivalent, so are duals

Conditionals • Given p q • Converse q p • Inverse ~p ~q • Contrapositive ~q ~p

Arguments • If you insulted Bob then I’ll never speak to you again. You insulted Bob so I’ll never speak to you again. • Premise • Conclusion

Arguments • If you are a mathematician then you are clever. You are clever and rich. Therefore If you are rich then you are a mathematician. • Premise • Conclustion