It's not entirely unlikely that I am remembering just the good stuff :) But I was surprised how many books would define even the most common notation, like ⊂, ∀, and ∃.

I guess if you call your book "Introduction to..." you ought to do that. And it seems that all books were called that, regardless of how narrow and advanced the rest of the title was :)

Often books assume some prerequisites, the question here is the level of those prerequisites. Some books try to include all the necessary background, others assume a pre-existing base level of knowledge.

Different authors, different books, different audiences, and different contexts.