If and only if

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A B A \iff B
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Iff (if and only if) is a "biconditional statement". It means both conditions must hold (be true) for the statement to be true. It is often called a necessary and sufficient condition. Example:

  • "Madison will eat the fruit if and only if it is an apple" (equivalent to "Madison will eat the fruit if the fruit is an apple, and will eat no other fruit.)

This makes it clear that Madison will eat all and only those fruits that are apples. She will not leave any apple uneaten, and she will not eat any other type of fruit. That a given fruit is an apple is both a necessary and a sufficient condition for Madison to eat the fruit.