Contraposition
Pronunciation: /ˌkɒn trə pəˈzɪ ʃən/ ?
A contraposition is how the
truth values of two
statements relate to each other.^{[1]} 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 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 |
Cite this article as:
Contraposition. 2009-12-21. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/contraposition.html.
Translations
Image Credits
Revision History
2009-12-21: Added "References" (
McAdams, David.)
2007-09-18: Initial version (
McAdams, David.)