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 isand the converse isthen the inverse isthe contrapositive isand the contradiction is
FalseFalseFalseFalseTrue
FalseTrueTrueFalseTrue
TrueFalseFalseTrueFalse
TrueTrueTrueTrueFalse

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.

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2009-12-21: Added "References" (McAdams, David.)
2007-09-18: Initial version (McAdams, David.)

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