Let A,B,C be subsets of a set. Prove that A ∩ B ⊆ C iff A⊆B' U C

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Suppose that $A\cap B\subseteq C$

$A=A\cap U=A\cap (B\cup B’)=(A\cap B)\cup (A\cap B’)$

We know that $A\cap B\subseteq C$ and $A\cap B’\subseteq B’$ .

It implies that $A=(A\cap B)\cup (A\cap B’)\subseteq B’\cup C$ .

So, $A\subseteq B’\cup C.$

Now suppose that $A\subseteq B’\cup C$ , (if $x ∈ A \text{, then}\ x ∈ B ’ ∪ C )$

If $x\in A\cap B,$ then $x\in B’\cup C$ and $x\in B$ .

So, $x\in (B’\cup C)\cap B=(B’\cap B)\cup (C\cap B)=C\cap B\subseteq C$ .

Therefore, $A\cap B\subseteq C.$