Simple Set Theory Question 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.
 A: 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 $C \subset B$.
A: We begin in the definitions.


*

*$A\cup C\subseteq A\cup B$ means that if $x\in A$ or $x\in C$ then $x\in A$ or $x\in B$.

*$A\cap C\subseteq A\cap B$ means that if $x\in A$ and $x\in C$ then $x\in A$ and $x\in B$.

*Lastly, $C\subseteq B$ which is what we want to show, namely if $x\in C$ then $x\in B$.
Now assume the first two happen. To show the last we need to show that what is stated in the definition is true. So we begin by taking some arbitrary $c\in C$ and going through what we already know:


*

*Suppose that $c\in C$. 

*Either $c\in A$ or $c\notin A$. 


*

*If $c\in A$ then $c\in A\cap C$, by the second statement we have that $c\in A\cap B$. In particular $c\in B$.

*If $c\notin A$ then $c\in A\cup C$, since $c\in C$, and by the first statement we have that $c\in A\cup B$. Therefore $c\in A$ or $c\in B$. However we assume that $c\notin A$ so we have to have $c\in B$.


*We showed that either way $c\in C$ implies that $c\in B$. Therefore $C\subseteq B$.

One tip I always repeat when teaching mathematics (especially at intro level) is to have the definitions of all the symbols and terms at reach. Most of these exercises can be solved by simply unraveling the definitions to their bare form and then reconstructing what we have (with the assumptions) to have the conclusion.
When approaching a problem the first thing which one needs to be able to do is tell exactly what each symbol mean and what are the relations between the different symbols. This is often ignored by most first year students. You cannot solve a problem that you do not understand, not in a meaningful and helpful way, anyway.
A: Suppose $x \in C$. You need to show $x \in B$. Since $x \in C$, $x \in A \cup C$, so either $x \in A$ or $x \notin A$. We consider two cases:
Case 1: $x \in A$. Then, since $x \in C$, we must have $x \in A \cap C$. What does this imply?
Case 2: $x \notin A$.Then $x \in C$. But $A \cup C \subset A \cup B$. What does this imply?  
