Thinking Mathematically Logic 3.5 Equivalent Statements and Variation of Conditional Statements

Equivalent Statements Equivalent compound statements aer made up of the same simple statements and have the same corresponding truth values for all true-false combinations of these simple statements.

Example: Equivalent Statements Exercise Set 3.5 #7 (p ^ q) ^ r p ^ (q ^ r)

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.

Example: Contrapositive Exercise Set 3.5 #19 What is the contrapositive of If I am in Chicago, then I am in Illinois.

Converse and Inverse The converse and inverse are contrapositives (of each other) and are equivalent. They are not equivalent to the original conditional statement. q  p is the converse of p  q. ~p  ~q is the inverse of p  q.

Example: Converse and Inverse Exercise Set 3.5 #19, 21 –What is the converse/inverse of If I am in Chicago, then I am in Illinois. –What is the converse/inverse/contrapositive of If the stereo is playing, then I cannot hear you.

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

Thinking Mathematically Logic 3.5 Equivalent Statements, Conditional Statements, and De Morgan’s Laws