equivalence classes, especially in category theory, where the equivalence relation is (categorical) isomorphism; for example, "The tensor product in a weak monoidal category is associative and unital up to a natural isomorphism." vanish To assume the value 0. For example, "The function sin(x) vanishes for those values of x that are integer multiples of π." This can also apply to limits: see Vanish at infinity. weak, weaker The converse of strong. well-defined Accurately and precisely described or specified. For example, sometimes a definition relies on a choice of some object; the result of the definition must then be independent of this choice. Proof terminology The formal language of proof draws repeatedly from a small pool of ideas, many of which are invoked through various lexical shorthands in practice. aliter An obsolescent term which is used to announce to the reader an alternative method, or proof of a result. In a proof, it therefore flags a piece of reasoning that is superfluous from a logical point of view, but has some other interest. by way of contradiction (BWOC), or "for, if not, ..." The rhetorical prelude to a proof by contradiction, preceding the negation of the statement to be proved. if and only if (iff) An abbreviation for logical equivalence of statements. in general In the context of proofs, this phrase is often seen in induction arguments when passing from the base case to the induction step, and similarly, in the definition of sequences whose first few terms are exhibited as examples of the formula giving every term of the sequence. necessary and sufficient A minor variant on "if and only if"; "A is necessary (sufficient) for B" means "A if (only if) B". For example, "For a field K to be algebraically closed it is necessary and sufficient that it have no finite field extensions" means "K is algebraically closed if and only if it has no finite extensions". Often used in lists, as in "The following conditions are necessary and sufficient for a field to be algebraically closed...". need to show (NTS), required to prove (RTP), wish to show, want to show (WTS) Proofs sometimes proceed by enumerating several conditions whose satisfaction will together imply the desired theorem; thus, one needs to show just these statements. one and only one A statement of the existence and uniqueness of an object; the object exists, and furthermore, no other such object