Let $A, B, C$ be sets. Prove that if $B \subseteq C$, then $(A\cap B)\subseteq (A\cap C)$ Let $A, B, C$ be sets. Prove that if $B \subseteq C$, then $(A\cap B)\subseteq (A\cap C)$
I tried this question so I'll tell you guys what I have. Please correct me if I am wrong.
Assume $B \subseteq C$, then if $x$ is an element of $B$, $x$ must be an element of $C$.
Then if $A$ was added to set $B$ and $A$ was added to set $C$, every element in $A\cap B$ must belong to $A\cap C$. 
I'm not really sure how to write that mathematically or symbolically since I'm new to this
 A: We have $A \cap B \subset A$ and $A \cap B \subset B \subset C$. Consequently, if $x \in A \cap B$, we have $x \in A$ and $x\in C$, from which we have $x \in A \cap C$. Hence $A \cap B \subset A \cap C$.
A: $
\newcommand{\calc}{\begin{align} \quad &}
\newcommand{\calcop}[2]{\notag \\ #1 \;\;\; & \;\;\; \text{"#2"} \notag \\ \quad & }
\newcommand{\endcalc}{\notag \end{align}}
$Why not try to simplify $\;A \cap B \;\subseteq\; A \cap C\;$, by expanding the definitions, simplifying using the rules of logic, and working towards $\;B \subseteq C\;$?
$$\calc
A \cap B \;\subseteq\; A \cap C
\calcop{\equiv}{definition of $\;\subseteq\;$}
\langle \forall x :: x \in A \cap B \;\Rightarrow\; x \in A \cap C \rangle
\calcop{\equiv}{definition of $\;\cap\;$, twice}
\langle \forall x :: x \in A \land x \in B \;\Rightarrow\; x \in A \land x \in C \rangle
\calcop{\equiv}{logic: remove superfluous part of consequent}
\langle \forall x :: x \in A \land x \in B \;\Rightarrow\; x \in C \rangle
\calcop{\Leftarrow}{logic: strengthen by weakening antecedent -- $\;A\;$ is not in our goal}
\langle \forall x :: x \in B \;\Rightarrow\; x \in C \rangle
\calcop{\equiv}{definition of $\;\subseteq\;$}
B \subseteq C
\endcalc$$
A: Your restatement of the meaning of the assumption $B \subseteq C$ is correct.
However, your proof that $A \cap B \subseteq A \cap C$ is incorrect. It really seems that you're just repeating the desired conclusion without justifying it.
To prove that a set $E$ is contained in a set $F$, take an element $x \in E$, and try to prove that $x$ must also belong to $F$. What sets $E$ and $F$ would you want to apply this idea to here?
