TITLE: Use of "without loss of generality"
QUESTION [73 upvotes]: Why do we use "without loss of generality" when writing proofs?
Is it necessary or convention? What "synonym" can be used?

REPLY [8 votes]: It means we're going to save time by appealing to symmetry. For instance, if $p$ is an irreducible element of a ring and $ab=p$, then precisely one of $a$, $b$ are units. Instead of considering these two cases separately, we might say: "Without loss of generality, assume $a$ is a unit." This is the same as saying: "We will consider the case where $a$ is a unit; the case where $b$ is a unit is similar."