Logic Worksheet With Answers Page 14
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22Equivalent compound statements are made up of the same simple statements and have the
same corresponding truth values for all true-false combinations of these simple statements.
If a compound statement is true, then its equivalent statement must also be true.
If a compound statement is false, its equivalent statement must also be false.
Using the truth tables, the corresponding columns for the two statements must be identical.
The symbol which is used to show an equivalence is .
Ex: Show that p ~q and ~p ~q are equivalent.
First we construct a truth table and see if the corresponding truth values are the same:
p ~q
~p ~q
p q
~q
~p
T T
F
T
F
T
T F
T
T
F
T
F T
F
F
T
F
F F
T
T
T
T
The two shaded columns are identical, so the statements are equivalent. We write this p ~q ~p
~q.
Ex: Show that p q
~q
~p.
~q ~p
p
q
p q
~p
~q
T T
T
F
F
T
F
T F
F
T
F
T
F T
T
F
T
F F
T
T
T
T
The two shaded columns are identical, so the statements are equivalent
The statement ~q
~p is called the contrapositive of the conditional p q. They are logical
equivalent.
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