There's a matrix cookbook. Is there a set cookbook?
I'm looking for a comprehensive list of properties, formulas, inclusions or lack thereof on the algebra of sets, De Morgan's laws, Cartesian products, infinite (countable or uncountable) intersections or unions, etc.
For example, in topology, real analysis, probability theory or measure theory, I see a lot of expressions like $$\bigcup_{k=1}^{\infty} (A_k \cap B_k)$$ $$ (A \cup B) \times (C \cup D)$$ $$ \bigcap_{x \in X} (A_x \times B_x)^c$$
Sometimes I wonder if we can do things like put/remove a $\cap B$ or $\cup A$ to both sides of a set inclusion. If it's a set equality, I think there are cases where it's a no longer equality but merely an inclusion or sometimes not an inclusion at all.
Expressions similar to the ones aforementioned don't always equal to something, they're usually just subsets or supersets of something else, and sometimes there's no (or no relevant) set inclusion. These are not difficult to prove; I'd just like to have a reference much like the matrix cookbook please.
I'm interested mainly in the basics and not necessarily about open/closed, non-/measurable, counterexamples, etc. If a set inclusion is true and the reverse doesn't hold, I'll take the cookbook's word for it.
I think this can be found in some appendix of some textbook. For example, a Calculus textbook appendix might have a list of trigonometric identities, without proofs, counterexamples or exercises. That's what I'm looking for.
Wiki is helpful but doesn't have a lot of the properties. For example, the Cartesian products page doesn't have the second expression above (other than explaining what the second expression is not) and whether or not an empty Cartesian product implies one of the Cartesian factors is empty, but I was able to find both and more on stackexchange.
There are books on these, but a lot of the properties are in exercises, some of which are determining true/false rather than dis/proving, so I'd have to try them myself or look up the solutions manual.
I wish there was some compilation of all these folklores.
The book " Naive Set Theory " by Paul Halmos is very informative and readable book.