Can you conclude that A = B if A, B, and C are sets such that...

Question

a. A ∪ C = B ∪ C

b. A ∩ C = B ∩ C

c. A ∩ C = B ∩ C and A ∪ C = B ∪ C

My method of solving this was to convert everything to propositional logic, then to solve it to show that none of the above are tautologies, therefore, my answer was no for all of them. However, I just wanted to make sure that my method works correctly, and if there is an easier way to solve these kinds of problems or not.

Answer 1

If a statement is true, you have to prove it; if it is false then you have to disprove it using an example.

Note that true propositions (in this case) are tautologies, but they include conditionals. That is, the last one which is true is in fact the following statement:

$$(A\cap C=B\cap C\land A\cup C=B\cup C)\rightarrow A=B$$

HINTS:

1. Consider the case where $A$ and $B$ are subsets of $C$.
2. Consider the case where $A$ and $B$ are disjoint from $C$.
3. This one you can prove by noting that $X\cup Y=(X\setminus Y)\cup Y$, and the latter is a disjoint union, and applying this several times to the various equations that you are given.