Contrapositive
The contrapositive of a conditional statement is formed by negating both the hypothesis and the conclusion, and then interchanging the resulting negations.
In other words, the contrapositive negates and switches the parts of the sentence. It does BOTH the jobs of the INVERSE and the CONVERSE.
Example:
Conditional: “If 9 is an odd number, then 9 is divisible by 2.”
This statement is logically FALSE.
Contrapositive: “If 9 is not divisible by 2, then 9 is not an odd number.”
This statement is logically FALSE.
HINT: Remember that the contrapositive (a big long word) is really the combining together of the strategies of two other words: converse and inverse.
An important fact to remember about the contrapositive, is that it always has the SAME truth value as the original conditional statement.
If the original statement is TRUE, the contrapositive is TRUE.
If the original statement is FALSE, the contrapositive is FALSE.
They are said to be logically equivalent.
(“equivalent” means “the same”)
A truth table can be used to show that a conditional statement and its contrapositive are logically equivalent. Notice that the truth values are the same.
A truth table is a pictorial representation of all of the possible outcomes of the truth value of a compound sentence. Letters such as and are used to represent the facts (or sentences) within the compound sentence.
| Conditional | Contrapositive | ||
| p | q | p→q | ∼q→∼p |
| T | T | T | T |
| T | F | F | F |
| F | T | T | T |
| F | F | T | T |
Remember:
The contrapositive is the mixing of the inverse and the converse.