Contraposition

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For contraposition in the field of traditional logic, see Contraposition (traditional logic).

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".

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".

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

The contradiction is "There exists a house that is not a building".

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.

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.

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.

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.Template:Logic-stub

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