Cfrgtkky

A particular question states: Show if $A ⊆ B^c$ then $A ∩ B = ∅$.
Being very new to set theory, I attempted to start some proof, which appears below, where $S =$ universe of discourse:
$$
A cap B = Acap(Ssetminus B^c) = (Acap S)setminus B^c = Asetminus B^c = emptyset.
$$

This may or may not be correct; I have no solution personally offered to this question. But I am wondering if it was permitted to "slide" the parentheses from $Acap (Ssetminus B^c) $ to $Acap B = (A cap S)setminus B^c $.

Usually if you need to ask if a step is permitted, then the answer is "no, unless..."—what comes next is either:

• ...it is part of the definition of something involved; or


• ...it invokes a result that has already been proved.

In this case, you want to know whether you can apply the rule $A cap (B setminus C) = (A cap B) setminus C$. It turns out that this rule is true in general, but if you wanted to use it then you'd need to prove it, or you'd need to cite somewhere that it is proved. I suspect that this isn't a result that you can use without justification, so you'd need to prove this too.

Be careful though! For example, it is not true in general that $A cup (B setminus C) = (A cup B) setminus C$, even though this looks very similar.

Coming back to the problem at hand: when you're new to set theory, the temptation in answering questions like this is to rearrange equations until you get the answer. I would advise against this, as it often leads to mistakes and increases the risk of invoking illegal 'rules'. Instead, to prove two sets are equal, you should prove that they have the same elements.

So for your question: assume that $A subseteq B^c$. To prove that $A cap B = varnothing$ you need to show that $A cap B$ and $varnothing$ have the same elements. Since $varnothing$ has no elements, this amounts to assuming that there is some $x in A cap B$ and then deriving a contradiction. This, in turn, is almost immediate from the assumption that $A subseteq B^c$.