"

**T**he **F**ollowing **A**re **E**quivalent": an

abbreviation often used in writing

Mathematics (especially on a

blackboard!).

Many times you have several "good" ways of defining the same thing. Which do you choose? It doesn't really matter, because the very next thing you'll find after the definition is:

**Theorem.** *TFAE*:
*Original definition*
*First restatement*
*Second restatement*...

So you could equally well have defined using (b) or (c) or ...

The claim of the theorem is that if *any one* of the conditions holds, then they *all* do.

To *prove* such a theorem, you don't have to show "(a) iff (b)" AND "(a) iff (c)"; it's enough to show that "if (a) then (b); if (b) then (c); if (c) then (a)". When you have many equivalent conditions, this saves a great deal of work.