# Contraposition

Pronunciation: /ˌkɒn.trə.pəˈzɪ.ʃən/ Explain

A contraposition is how the truth values of two statements relate to each other. Given a statement 'if p then q' where p and q are two related statements:

• The converse is 'if q then p';
• The inverse is 'if not p then not q';
• The contrapositive is 'if not q then not p'; and
• The contradiction is 'there exists an example of p for which q is false.

#### Example 1

If a shape is a square then it is also a rectangle. This statement is true.

• Converse: If a shape is a rectangle then the shape is a square. This statement is false.
• Inverse: If a shape is not a square then the shape is not rectangle. This statement is false.
• Contrapositive: If a shape is not a rectangle then the shape is not a square. This statement is true.
• Contradiction: There exists a square that is not a rectangle. This statement is false.

#### Example 2

In a triangle, if two angles are equal then the sides opposite the equal angles are equal. This statement is true.

• Converse: In a triangle, if the sides opposite two angles are equal then the two angles are equal. This statement is also true.
• Inverse: In a triangle, if two angles are not equal then the sides opposite two angles are not equal. This statement is true.
• Contrapositive: In a triangle, if the sides opposite two angles are not equal then two angles are not equal. This statement is true.
• Contradiction: There exists a triangle where two angles are equal and the sides opposite the equal angles are not equal. This statement is false.

#### Example 3

If a shape is a square then it is a circle. This statement is false.

• Converse: If a shape is a circle then the shape is a square. This statement is false.
• Inverse: If a shape is not a square then the shape is not circle. This statement is false.
• Contrapositive: If a shape is not a circle then the shape is not a square. This statement is false.
• Contradiction: There exists a square that is not a circle. This statement is true.

#### Truth Table for Contraposition

 If a statement is and the converse is then the inverse is the contrapositive is and the contradiction is False False False False True False True True False True True False False True False True True True True False
1. McAdams, David E.. All Math Words Dictionary, contraposition. 2nd Classroom edition 20150108-4799968. pg 45. Life is a Story Problem LLC. January 8, 2015. Buy the book

McAdams, David E. Contraposition. 4/16/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/c/contraposition.html.

### Revision History

4/16/2019: Updated equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)