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, having been supplanted by the Halmos end-of-proof mark, a square sign ∎. 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. without (any) loss of generality (WLOG, WOLOG, WALOG), we may assume (WMA) 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. Proof techniques Mathematicians have several phrases to describe proofs or proof techniques. These are often used as hints for filling in tedious details. angle chasing Used to describe a geometrical proof that involves finding relationships between the various angles in a diagram. back-of-the-envelope calculation An informal computation omitting much rigor without sacrificing correctness. Often this computation is "proof of concept" and treats only an accessible special case. brute force Rather than finding underlying principles or patterns, this is a method where one