I'm starting to learn Set Theory and I'm stuck on a question:
Show that the relations $$(A \cup C)\subset(A\cup B), (A\cap C) \subset (A\cap B)$$
when combined, imply $C\subset B$. If it's in anyone's interest, this is from the online textbook "Basic Concepts of Mathematics" by Elias Zakon. I'm afraid I've no idea where to start. Any help would be much appreciated.
[Math] Simple Set Theory Question
elementary-set-theory
Best Answer
Suppose $c$ is in $C$. We want to show that $c$ is in $B$. Certainly $C$ is in $A \cup C$, and so by your first assumption, $c$ is in $A \cup B$. That is, either $c$ is in $A$ or $c$ is in $B$. In the latter case we are done. In the former case, $c$ is in $C$ and in $A$ and so $c$ is in $A \cap C$, and so by your second assumption, $c$ is in $A \cap B$ and hence in $B$.
Thus in all cases, if $c$ is in $C$, then $c$ is in $B$, and so we have shown $C \subset B$.