Question

how to prove this equation A∩B⊂A⊂A⊂B

Answer 1

Conditions

how to prove this equation $A \cap B \subset A \subset A \subset B$

Solution

First of all there should be a little mistake, I think the correct conditions is how to prove:

$$(A \cap B) \subset A \subset (A \cup B)$$

Let's prove first:

$$(A \cap B) \subset A$$

As we know, the elements from $(A \cap B)$ are those elements, which are belong as to $A$ as to $B$ .

That's why each of these elements is in A, but if we take some element from A, but which is not in B, so it will not be in $(A \cap B)$ . That's why $(A \cap B) \subset A$ .

Now let's prove:

$$A \subset (A \cup B)$$

As we know, the elements from $(\mathbf{A} \cup \mathbf{B})$ are those elements, which are belong to A or to B. Consider 2 sets A and B, which have not common points. For any element of A, it is belong to $(\mathbf{A} \cup \mathbf{B})$ , but let's take an element from B, it's not in A, but in $(\mathbf{A} \cup \mathbf{B})$ . That's why $\mathbf{A} \subset (\mathbf{A} \cup \mathbf{B})$

Q.E.D.