Explanation: What does WLOG mean?

(related to Part: Methods of Mathematical Proving)

Proving mathematical theorems becomes sometimes technically unnecessarily complicated if one tries to make too general assumptions in the proof. In order to avoid such technicalities, mathematicians quite often make restrictions on these assumptions and narrow the generality of their assumptions concentrating on some special cases, and making the whole proof easier to follow.

When restricting assumptions in mathematical proofs, mathematicians use the phrase "without loss of generality", which sometimes is abbreviated by WLOG. What does this phrase mean?

It means that even though the restriction is being placed on one of the assumptions made in the proof, if one can complete the proof for this special case, then it would be very easy to give a proof for similar, but different special cases, exhausting all cases necessary to prove the general case.

For instance, imagine that the assumption for a proof is, that we are given some real numbers $a,b\in\mathbb R$ such that $a\neq b,$ and that it is easier to prove a theorem following from the inequality $a\neq b$ that, in addition, we restrict this inequality to the special case $a < b.$ If the reasoning in the proof does not depend on this special case and works also for $b < a$, while exchanging the denotations $a$ and $b$ in the whole proof, mathematicians could begin their proof with the following phrases:

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  1. Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016