Equivalent Statements 3.4 Equivalent Statements

Equivalent Statements Two statements are equivalent if both statements have exactly the same truth values in the answer columns of the truth tables. In a truth table, if the answer columns are identical, the statements are equivalent. If the answer columns are not identical, the statements are not equivalent.

De Morgan’s Laws

Example: Using De Morgan’s Laws to Write an Equivalent Statement Use De Morgan’s laws to write a statement logically equivalent to “Benjamin Franklin was not a U.S. president, but he signed the Declaration of Independence.” Solution: Let p: Benjamin Franklin was a U.S. president q: Benjamin Franklin signed the Declaration of Independence The statement symbolically is ~p Λ q.

Example: Using De Morgan’s Laws to Write an Equivalent Statement The logically equivalent statement in symbolic form is Therefore, the logically equivalent statement to the given statement is: “It is false that Benjamin Franklin was a U.S. president or Benjamin Franklin did not sign the Declaration of Independence.”

To change a conditional statement into a disjunction, negate the antecedent, change the conditional symbol to a disjunction symbol, and keep the consequent the same. To change a disjunction statement to a conditional statement, negate the first statement, change the disjunction symbol to a conditional symbol, and keep the second statement the same.

Variations of the Conditional Statement The variations of conditional statements are the converse of the conditional, the inverse of the conditional, and the contrapositive of the conditional.

Variations of the Conditional Statement “if not q, then not p” ~p ~q Contrapositive of the conditional “if not p, then not q” Inverse of the conditional “if q, then p” p q Converse of the conditional “if p, then q” Conditional Read Symbolic Form Name

Homework P. 136 # 9 – 71 eoo