Thinking Mathematically Equivalent Statements, Conditional Statements, and De Morgan’s Laws

Equivalent Statements Two statements are “equivalent” if they have the same truth values.

A Conditional Statement and Its Equivalent Contrapositive p  q ≡ ~q  ~p The truth value of a conditional statement does not change if the antecedent and consequent are reversed and both are negated. The statement ~q  ~p is called the contrapositive of the conditional p  q.

Converse and Inverse The converse and inverse are contrapositives and are equivalent. q  p is the converse of p  q. ~p  ~q is the inverse of p  q.

Conditional Statements Let p and q be statements. NameSymbolic Form Conditional p  q Converseq  p Inverse~p  ~q Contrapositive~q  ~p

The Negation of a Conditional Statement The negation of p  q is p /\ ~q. This can be expressed as ~(p  q) ≡ p /\ ~q.

De Morgans Laws ~(p/\q) ≡ ~p\/~q

“De Morgans Laws” ~(p\/q) ≡ ~p/\~q