2
$\begingroup$

I'm trying to prove the following statement:

$$A\cap B \subseteq C \iff A \subseteq \overline{B} \cup C$$

I need to do it using a formal proof.. I've tried to do it for some time now and couldn't find anything close..

Thanks!

$\endgroup$

migrated from mathematica.stackexchange.com May 5 '15 at 12:04

This question came from our site for users of Wolfram Mathematica.

2
$\begingroup$

$\Rightarrow$:

$$ A = \underbrace{(A\cap B)}_{\subseteq C} \cup \underbrace{(A\cap\overline{B})}_{\subseteq \overline{B}} \subseteq C\cup \overline{B}$$

$\Leftarrow$:

$$A\cap B \subseteq (\overline{B}\cup C)\cap B = (\overline{B}\cap B)\cup (C\cap B) = C\cap B\subseteq C$$

$\endgroup$
0
$\begingroup$

I suppose that $\bar B$ is the complementary of $B$.

Suppose $A\cap B\subset C$ and let $x\in A$. If $x\in A\cap B$, by hypothesis, $x\in C$ and thus $x\in \bar B\cup C$. If $x\notin A\cap B$, then $x\in A\cap \bar B$ and thus $x\in \bar B$. Therefore $x\in \bar B\cup C$.

Reciprocally, suppose that $A\subset \bar B\cup C$ and let $x\in A\cap B$. In particular, $x\in A$ and $x\in B$. By hypothesis, $x\in \bar B\cup C$. If $x\in \bar B$, we have a contradiction with $x\in B$, therefore $x\in C$, what prove the claim.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.