Draw a Venn diagram for: (A $\subseteq$ B) $\Rightarrow$ C $\cap$ A $\subseteq$ C $\cap$ B

Question

Let M be an arbitrary set and let A, B, C ⊆ M be subsets of M. Prove or use a counterexample to disprove the validity of the following statement:

(A $\subseteq$ B) $\Rightarrow$ C $\cap$ A $\subseteq$ C $\cap$ B

I don’t know where to start with this kind of “subset/implies” statements and I can’t find something similar in textbooks/online.

The statement is valid.

I’m trying to draw a Venn diagram to see for myself first if the statment is valid or not, before proceeding further. But I’m not succeeding at visualizing it.

I’m looking for inputs on how it should look like.

Thanks!

Answer 1

Since $A \subset B$, any point of $A$ lays in $B$. Since any point of $A \cap C$ lays in $A$ and in $C$, any point of $A\cap C$ is also inside $B$ and $C$, which means its in $B \cap C$.