Contraposition is the concept of how two qualities or statements relate to each other. In mathematics, for the statement "if P, then Q" for any two propositions P and Q, the converse is "if Q, then P", the inverse is "if not P, then not Q", and the contrapositive is "if not Q, then not P". Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... This article is about the word proposition as it is used in logic, philosophy, and linguistics. ...

Example

Take the statement "All houses are buildings". This can be equivalently expressed as "If an object is a house, then it is a building".

The converse is "If an object is a building, then it is a house".

The inverse is "If an object is not a house, then it is not a building". In logic, if S is a statement of the form P implies Q then the inverse of S is a statement of the form (not P) implies (not Q). ...

The contrapositive is "If an object is not a building, then it is not a house".

The contradiction is "There is a house that is not a building". Broadly speaking, a contradiction is an incompatibility between two or more statements, ideas, or actions. ...

Truth

• If a statement is true, then its contrapositive is always true (and vice versa).
• If a statement is false, its contrapositive is always false (and vice versa).
• If a statement's inverse is true, its converse is always true (and vice versa).
• If a statement's inverse is false, its converse is always false (and vice versa).
• If a statement's contradiction is false, then the statement is true.
• If a statement (or its contrapositive) and the inverse (or the converse) are both true or both false, it is known as a logical biconditional.

In logical calculus of mathematics, logical biconditional is a logical operator connecting two statements to assert, p if and only if q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ...

Application

Because the contrapositive of a statement always has the same truth value (truth or falsity) as the statement itself, it can be a powerful tool for proving mathematical theorems via proof by contradiction, as in the proof of the irrationality of the square root of 2. In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ... Look up theorem in Wiktionary, the free dictionary. ... This article or section does not cite any references or sources. ... In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers. ...

By the definition of a rational number, the statement can be made that "If is rational, then it can be expressed as an irreducible fraction". This statement is true because it is a restatement of a true definition. In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... An irreducible fraction (or fraction in lowest terms) is a vulgar fraction in which the numerator and denominator are smaller than those in any other equivalent fraction. ...

The contrapositive of this statement is "If cannot be expressed as an irreducible fraction, then it is not rational". This contrapositive, like the original statement, is also true. Therefore, if it can be proven that cannot be expressed as an irreducible fraction, then it must be the case that is not a rational number.