Prove that $A - C = B - C$ iff $A \cup C = B \cup C$.
I tried to show $A \cup C = B \cup C$ from $A - C = B - C$ using the absorption law:
$ A \cup C \\ = A \cup (A - C) \cup C \\ = A \cup (B - C) \cup C \\ = A \cup (B \cap C') \cup C \\ = A \cup B \cup C $
What's wrong?
On
This is a case where "proof by venn diagram" works well.
So if $A\cup C=B\cup C$, you can see that there can't be anything in just $A$ or just $C$, ie zones $100, 010$. From that you get that $A-C=B-C=110$.
If $A-C=B-C$ we have $100,110=110,010$, so once again we have $100$ and $010$ are empty, thus its an if and only if.
You can formalize this by breaking down each region into intersections of either $A,B,C$ or $A^C,B^C,C^C$, IE $100=A\cap B^C\cap C^C$
There is nothing wrong with what you wrote, but it just doesn't quite prove what you want.
You can however do the same set of steps with $A$ and $B$ swapped to show that $B\cup C=B\cup A \cup C$. You can then combine the two results to get
$$A\cup C=A\cup B \cup C =B\cup A \cup C = B\cup C$$
Note that this only proves one direction of the iff statement that you need to prove.