Mathematics: If a ⊂ b, then a ∩ b = a ∪ b. always sometimes or never

If a ⊂ b, then a ∩ b = a ∪ b. always sometimes

Ответ:

ayoismeisalex

10.03.2022

Mathematics

Sometimes. Proof:

Assume $A\subset B$ .

Let $a\in A\cap B$ , so that $a\in A$ and $a\in B$ . Then of course $a\in A\cup B$ , so $A\cap B\subseteq A\cup B$ .

Now let $a\in A\cup B$ , so that $a\in A$ *or* $a\in B$ . Since $A\subset B$ , if $a\in A$ then $a\in B$ , so that $a\in A\cap B$ . But if $a\in B$ , then $a\not\in A$ , so it's not always the case that $a\in A\cap B$ , and therefore in general $A\cup B\not\subseteq A\cap B$ .