``Iff'' is a shorthand for the phrase ``if and only if.'' This phrase is used in assertions. If I assert that ``A if and only if B'' I mean that A is true if B is true, and furthermore, A is true only when B is true. If I say ``C if D'' then I know nothing about C when D is false, or about D when C is true. Similarly, were I to say ``C only if D'' then I am telling you nothing about C when D is true, or about D when C is false. Got it??
You can say the same thing in a variety of ways. Saying ``A iff B'' is the same thing as saying ``A implies B, and B implies A,'' or, one of my favorites, that ``A is a necessary and sufficient condition for B,'' or, equivalently, ``B is a necessary and sufficient condition for A.''
The following are equivalent conditions:
So if I say that A is necessary and sufficient for B, then we get: A if and only if B,
and (B or (not A)) and (A or (not B)) which is equivalent to ((A and B) or ((not A) and (not B))) -- in other words, A and B are either both true or both false. A and B are equivalent.