If C is a subset of A and B, then it is a subset of their intersection

by RoRi

June 20, 2015

Prove that if $C \subseteq A$ and $C \subseteq B$ , then $C \subseteq A \cap B$ .

Proof. Let $x$ be any element in $C$ . Then, $x \in A$ since $C \subseteq A$ , and $x \in B$ since $B \subseteq C$ . Therefore, $x \in A \cap B$ . Hence, $C \subseteq A \cap B$ .∎

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June 30, 2020 at 9:34 am

Anonymous says:

I am confident you meant to say $C \subset B$ instead of $B \subset C$ .
Subsequently the proof should look as follows:

$(C\subset A)\land (C \subset B) \iff (C \subset (A \cap B))$

Let $x$ be an element in $C$ . Then $x\in A$ since $C \subset A$ , and $x \in B$ since $C \subset B$ .
Therefore, $(x \in A)\land (x \in B) \iff x \in A \cap B$ . Hence, $C \subset A \cap B$ . $\blacksquare$

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