Converse, Inverse, Contrapositive
Given an if-then statement "if $p$ , then $q$ ," we can create three related statements:
A conditional statement consists of two parts, a hypothesis in the “if” clause and a conclusion in the “then” clause. For instance, “If it rains, then they cancel school.”
"It rains"
is the hypothesis.
"They cancel school"
is the conclusion.
To form the converse of the conditional statement, interchange the hypothesis and the conclusion.
The converse of
"If it rains, then they cancel school"
is
"If they cancel school, then it rains."
To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion.
The inverse of
“If it rains, then they cancel school”
is
“If it does not rain, then they do not cancel school.”
To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement.
The contrapositive of
"If it rains, then they cancel school"
is
"If they do not cancel school, then it does not rain."
Statement | If $p$ , then $q$ . |
Converse | If $q$ , then $p$ . |
Inverse | If not $p$ , then not $q$ . |
Contrapositive | If not $q$ , then not $p$ . |
If the statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true.
Example 1:
Statement | If two angles are congruent, then they have the same measure. |
Converse | If two angles have the same measure, then they are congruent. |
Inverse | If two angles are not congruent, then they do not have the same measure. |
Contrapositive | If two angles do not have the same measure, then they are not congruent. |
In the above example, since the hypothesis and conclusion are equivalent, all four statements are true. But this will not always be the case!
Example 2:
Statement | If a quadrilateral is a rectangle, then it has two pairs of parallel sides. |
Converse | If a quadrilateral has two pairs of parallel sides, then it is a rectangle. (FALSE!) |
Inverse | If a quadrilateral is not a rectangle, then it does not have two pairs of parallel sides. (FALSE!) |
Contrapositive | If a quadrilateral does not have two pairs of parallel sides, then it is not a rectangle. |