Give an example of two non-empty sets $A$ and $B$ such that {$A∪B$, $A∩B$, $A−B$, $B−A$} is the power set of some set. 
Give an example of two non-empty sets $A$ and $B$ such that {$A∪B$, $A∩B$, $A−B$, $B−A$} is the power set of some set.

This is problem 1.33 in Polimeni, Chartrand, and Zhang's Mathematical Proofs: A Transition to Advanced Mathematics. What is a good strategy for a problem of this kind? I understand what I am being asked to do; I am just unsure about the best way to begin.
 A: An answer is in the comments, but here is a way to get there. You know (maybe?) that the power set of a set with $n$ elements has cardinality $2^n$. Your set has cardinality $4$, so it should be the power set of a set with only $2$ elements.
Therefore you know that there can only be $2$ different elements in $A$ and $B$ (which is to say, $|A\cup B|=2$). From here you can just try a couple of different things until you get something that works. Challenge: can you find a different solution than the one in the comments?
EDIT. Actually, there can be other solutions if some or all of the four sets are equal. In fact, there are solutions with both $n=1$ and $n=0$ as well. Can you find them? I think it will be good practice.
A: In no particular order, here are some general steps that can help you to solve problems:

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*Get the definitions for everything you're working with, and make sure you're clear on their meaning

*Try some basic examples to get a feel for the problem. How we understand maths is weird. We may struggle to understand what a definition is telling us, but completely get it with a few examples — don't underestimate the utility of plugging things in

*Break the problem down into parts by asking questions. For instance, how many subsets can be made out of an arbitrary set? How many subsets are being created using the given instructions? Does the problem actually have a solution? Are there commonalities between this problem and other problems you've worked on before?

*Put all the information together. Lists work for some peeps, mind maps are good too, as are doodles, diagrams and annotations

If you're really struggling with a problem, here's some steps that can help:

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*Have a break and go and do something else. I find exercise, housework, watching nonsense, and showering to work for me. The idea is to de-stress and let your mind work on the problem in the background

*Explain the full problem — out loud — to your favourite toy. Assume that your toy doesn't know much about maths and everything has to be explained to them. Imagine questions that your interested toy would ask about your reasoning, and then answer them (this method is also useful for spotting errors in your writing). This is basically the Rubber Duck Method from programming

*Try to solve a simpler problem that's related, then see if you can build off it

*Ask for help

