Conditional and contrapositive
Converse and inverse
Practice makes perfect
We are looking for statements that are both true or both false. Two statements are logically equivalent when they have the same truth table. Let's list the truth table of a conditional statement, the converse, the inverse, and the contrapositive.
Conditional
p
q
p→ q
T
T
T
T
F
F
F
T
T
F
F
T
Converse
q
p
q→ p
T
T
T
T
F
T
F
T
F
F
F
T
Inverse
~ p
~ q
~ p → ~ q
F
F
T
F
T
T
T
F
F
T
T
T
Contrapositive
~ q
~ p
~ q → ~ p
F
F
T
T
F
F
F
T
T
T
T
T
Comparing the four tables, we see that the conditional and contrapositive have the same truth table, which means they are logically equivalent. The same applies to the converse and inverse. This means that if a conditional statement is true, then its contrapositive is necessarily true as well. The same logic applies for the converse and inverse.
