Prove that if B−C ⊆ $A^c$ then A∩B⊆C

Question

This is my proof:

let x ∈ B - C

which means: x ∈ B and x ∉ C and $A^c$ = B ∪ C

and because x ∈ B , B−C ⊆ $A^c$ .

and because x ∈ B and x ∉ A: A ∩ B = ∅ ⊆ C because ∅ is a subset of any set.

Did I prove this correctly?

Answer 1

Your proof starts with $x\in B-C$ and then $A^c = B\cup C$ which doesn't follow.

You're trying to show if $B-C\subset A^c$, then $A\cap B\subset C$. Here's an approach:

$B=(B\cap C)\cup (B-C)$, so $$A\cap B = A\cap((B\cap C)\cup (B-C)) = (A\cap (B\cap C))\cup (A\cap B-C)$$But $B-C\subset A^c$ by assumption, so $A\cap B-C\subset A\cap A^c = \emptyset$. Therefore, $$A\cap B = A\cap (B\cap C) = A\cap B \cap C\subset C$$

Alternatively, suppose $x\in A\cap B$ and $x\not\in C$, then $$x\in A\cap B\cap C^c = A\cap (B\cap C^c)\subset A\cap A^c = \emptyset$$Contradiction, so $A\cap B\subset C$