A very specialized form of a theorem begins with the statement “The following are equivalent…,” which is then followed by a list of statements. Informally, this lead-in sometimes gets abbreviated by “TFAE.” This formulation means that any two of the statements on the list can be connected with an “if and only if” to form a theorem. So if the list has n statements then, there are n(n−1)2 possible equivalences that can be constructed (and are claimed to be true).
Suppose a theorem of this form has statements denoted as A,B,C, …, Z. To prove the entire theorem, we can prove A⇒B,B⇒C,C⇒D, …, Y⇒Z and finally, Z⇒A. This circular chain of n equivalences would allow us, logically, if not practically, to form any one of the n(n−1)2 possible equivalences by chasing the equivalences around the circle as far as required.
