CHAPTER Special Kinds of Theorems 3 There are several kinds of theorems whose proofs follow rather standard structures. In this chapter we will look at some of the most important and common of these special theorems. "IF AND ONLY IF" OR "EQUIVALENCE THEOREMS" Statements including the expression "if and only if" are rather common and very useful in mathematics. If we can show that "A if and only if B," we are proving that A and B are (logically) equivalent statements, because either one of them is true (or false) only when the other one is true (or false). The statement "A if and only if B" means that "A is a necessary and sufficient condition for B" and that at the same time "B is a necessary and sufficient condition for A." Thus, to prove that the statement "A if and only if B" is true, we must prove two statements: