Question

For any two sets A and B, show that the following statements are equivalent :

(i) $A\subset B$ (ii) $A-B=\varphi$

(iii) $A\cup B=B$ (iv) $A\cap B=A.$

Answer 1

(i) In order to show that the following four statements are equivalent, we need to show that (1) $\Rightarrow$ (2), (2) $\Rightarrow$ (3), (3) $\Rightarrow$ (4) and (4) $\Rightarrow$ (1)

We first show that (1) $\Rightarrow$ (2)

We assume that $A\subset B$, and use this to show that $A-B=\varphi$

Now A - B = {$xϵA:x\text{/}ϵB$}, As $A\subset B$,

$\therefore$ Each element of A is an element of B,

$\therefore$ A - B = $\varphi$

Hence, we have proved that (1) $\Rightarrow$ (2).

(ii) We now show that (2) $\Rightarrow$ (3)

So assume that A - B = $\varphi$

To show $A\cup B=B$

A - B = $\varphi$

Every element of A is an element of B

[$\therefore A-B=\varphi$ only when there is some element in A which is not in B]

So $A\subset B$ and therefore $A\cup B=B$

So (2) $\Rightarrow$ (3) is true.

(iii) $A\cup B=B$

We now show that (3) $\Rightarrow$ (4)

Assume that $A\cup B=B$

To show : $A\cap B=A$

$\because A\cup B=B$

$\therefore A\subset BandsoA\cap B=A$

So (3) $\Rightarrow$ (4) is true.

(iv) $A\cap B=A$

Finally, we show that (4) $\Rightarrow$ (1), which will prove the equivalence of the four statements.

So, assume that $A\cap B=A$

To show : $A\subset B$

$\because A\cap B=A,$ thereore $A\subset B$ and

so (4) $\Rightarrow$ (1) is true.

Hence, (1) $\Rightarrow$ (2) $\iff$ (3) $\iff$ (4).