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Mathematics vocabulary · Sets and Logic

If and only if

Meaning in mathematics, with examples

phraseCommon term

In mathematics, if and only if means: A phrase joining two statements that are either both true or both false, so that each one implies the other; written ⇔ or 'iff'.

In Turkish: ancak ve ancak, gerek ve yeter koşul

How it is used

Common in definitions and theorems: 'n is even if and only if n² is even'. To prove such a statement you prove both directions. In writing it is often shortened to 'iff', but it is still read aloud as 'if and only if'.

Example sentences

  1. A triangle is equilateral if and only if all its angles are 60°.

  2. To prove an if and only if statement, prove each direction separately.

Common collocations

  • if and only if
  • iff
  • necessary and sufficient
  • prove both directions
  • converseThe statement you get by swapping the 'if' part and the 'then' part of a conditional; the converse of 'if p, then q' is 'if q, then p'.
  • proofA logical argument that shows, step by step from accepted facts, that a mathematical statement is always true.
  • theoremAn important mathematical statement that has been proved to be true, such as the Pythagorean theorem.
  • it follows thatA linking phrase used in proofs and solutions to show that the next statement is a logical result of what has just been shown.

More terms: Sets and Logic

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