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 |
References
- 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
Cite this article as:
McAdams, David E. Contraposition. 12/21/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/contraposition.html.
Revision History
12/21/2018: Reviewed and corrected IPA pronunication. (
McAdams, David E.)
6/25/2018: Removed broken links, updated license, implemented new markup, updated GeoGebra apps. (
McAdams, David E.)
12/21/2009: Added "References". (
McAdams, David E.)
9/18/2007: Initial version. (
McAdams, David E.)