List of Mathematical Jargon - Proof Terminology

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"; necessary means "only if" and sufficient means '"if". 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

An statement of the uniqueness of an object; the object exists, and furthermore, no other such object exists.

Q.E.D.

(Quod erat demonstrandum): A Latin abbreviation, meaning "which was to be demonstrated", historically placed at the end of proofs, but less common currently.

sufficiently nice

A condition on objects in the scope of the discussion, to be specified later, that will guarantee that some stated property holds for them. When working out a theorem, the use of this expression in the statement of the theorem indicates that the conditions involved may be not yet known to the speaker, and that the intent is to collect the conditions that will be found to be needed in order for the proof of the theorem to go through.

the following are equivalent (TFAE)

Often several equivalent conditions (especially for a definition, such as normal subgroup) are equally useful in practice; one introduces a theorem stating an equivalence of more than two statements with TFAE.

transport of structure

It is often the case that two objects are shown to be equivalent in some way, and that one of them is endowed with additional structure. Using the equivalence, we may define such a structure on the second object as well, via transport of structure. For example, any two vector spaces of the same dimension are isomorphic; if one of them is given an inner product and if we fix a particular isomorphism, then we may define an inner product on the other space by factoring through the isomorphism.

Let V be a finite-dimensional vector space over k....Let (e_i)_{1 ≤ i ≤ n} be a basis for V....There is an isomorphism of the polynomial algebra k_{1 ≤ i,j ≤ n} onto the algebra Sym_k(V ⊗ V*)....It extends to an isomorphism of k to the localized algebra Sym_k(V ⊗ V*)_D, where D = det(e_i ⊗ e_j*)....We write k for this last algebra. By transport of structure, we obtain a linear algebraic group GL(V) isomorphic to GL_n. —Igor Shafarevich (1991, p.12)

without (any) loss of generality (WLOG, WOLOG, WALOG), we may assume (WMA), it may be assumed that (WOLOGIMBAT)

Sometimes a proposition can be more easily proved with additional assumptions on the objects it concerns. If the proposition as stated follows from this modified one with a simple and minimal explanation (for example, if the remaining special cases are identical but for notation), then the modified assumptions are introduced with this phrase and the altered proposition is proved.