Q:

Let A,B,C and D be sets. Prove that

if $C ⊆ A$ and $ D ⊆ B$ , then $C ∩ D ⊆ A  ∩ B$.
How does one go about saying that this is true? I understand that if c is a subset of A that this means that there are elements of C in A. But what I do not get is what the elements are. I also understand that C is and Intersection of D but what I do not get is how do you prove this if you do not know what the variables sets are? So you cannot see what they have in common. Any advise on how to start would be good.

A:

Since $C \subseteq A$, if $x \in C$, then $x \in A$ (by the definition of subset).
Similarly, since $D \subseteq B$, if $x \in D$, then $x \in B$.
Now if $x \in C \cap D$, then by the definition of intersection, $x \in C$ and $x \in D$. From (1), since $x \in C$, we know that $x \in A$. From (2), since $x \in D$, we know that $x \in B$.
By the definition of intersection, if $x \in A$ and $x \in B$, then $x \in A \cap B$.
Therefore all $x \in C \cap D$ are also in $A \cap B$. So by the definition of subset, $C \cap D \subseteq A \cap B$.

