Question

4.
Show that the following four conditions are equivalent:
(i) A B(ii) A - B = 0 (iii) AUB=B (iv) An B = A​

Answer 1

Step-by-step explanation:

First, we have to show that (i) ⇔ (ii).

(i)⇒ (ii).

Let A⊂B

To show: A−B=ϕ

If possible, suppose A−B



=ϕ

This means that there exists x∈A,x∈

/

B, which is not possible as A⊂B.

∴A−B=ϕ

∴A⊂B⇒A−B=ϕ

(ii)⇒ (i).

Let A−B=ϕ

To show: A⊂B

Let xϵA

Clearly, xϵB because if x



ϵB, then A−B



=ϕ

∴A−B=ϕ⇒A⊂B

Hence, (ii)⇔(i)

(i)⇒ (iii).

Let A⊂B

To show: A∪B=B

Clearly, B⊂A∪B

Let xϵA∪B

⇒xϵA or xϵB

Case I: xϵA

⇒xϵB [∵A⊂B]

∴A∪B⊂B

Case II: xϵB

Then A∪B⊂B

So, A∪B=B

(iii)⇒ (i).

Conversely, let A∪B=B

To show : A⊂B

Let xϵA

⇒xϵA∪B [∵A⊂A∪B]

⇒xϵB [∵A∪B=B]

∴A⊂B

Hence, (iii)⇔(i)

Now, we have to show that (i)⇔(iv).

Let A⊂B

Clearly A∩B⊂A

Let xϵA

We have to show that xϵA∩B

As A⊂B,xϵB

∴xϵA∩B

∴A⊂A∩B

Hence, A=A∩B

Conversely, suppose A∩B=A

Let xϵA

⇒xϵA∩B

⇒xϵA and xϵB

⇒xϵB

∴A⊂B

Hence, (i)⇔(iv)