Proof Technique ME Multiple Equivalences
¶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 \(\tfrac{n(n-1)}{2}\) possible equivalences that can be constructed (and are claimed to be true).
Suppose a theorem of this form has statements denoted as \(A\text{,}\) \(B\text{,}\) \(C\text{,}\) …, \(Z\text{.}\) To prove the entire theorem, we can prove \(A\Rightarrow B\text{,}\) \(B\Rightarrow C\text{,}\) \(C\Rightarrow D\text{,}\) …, \(Y\Rightarrow Z\) and finally, \(Z\Rightarrow A\text{.}\) This circular chain of \(n\) equivalences would allow us, logically, if not practically, to form any one of the \(\tfrac{n(n-1)}{2}\) possible equivalences by chasing the equivalences around the circle as far as required.