If and only if

"Iff" redirects here. For other uses, see IFF (disambiguation).

INPUT		OUTPUT
A	B	A [math]\displaystyle{ \iff }[/math] B
0	0	1
0	1	0
1	0	0
1	1	1

In logic and mathematics, if and only if (sometimes abbreviated as iff) is a logical operator denoting a logical biconditional (often symbolized by [math]\displaystyle{ \leftrightarrow }[/math] or[math]\displaystyle{ \iff }[/math]). It is often used to conjoin two statements which are logically equivalent.

In general, given two statement A and B, the statement "A if and only if B" is true precisely when both A and B are true or both A and B are false. In which case, A can be thought of as the logical substitute of B (and vice versa).

An "if and only if" statement is also called a necessary and sufficient condition. For example:

• "Madison will eat the fruit if and only if it is an apple" is equivalent to saying that "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.

Note that the truth table shown is also equivalent to the XNOR gate.

If And Only If Media

• 

  A is a proper subset of B. A number is in A only if it is in B; a number is in B if it is in A.

• 

  C is a subset but not a proper subset of B. A number is in B if and only if it is in C, and a number is in C if and only if it is in B.

Related pages

• Equality (mathematics)
• Implication (logic)
• Propositional logic