Question

Q 5. For any two sets A and B, show that the following statements are equivalent:
(i) A is a subset of B
(ii) A-B= empty set
(iii) A union B= B
(iv) A intersection B= A

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Answer 1

Answer:

A-B

Step-by-step explanation:

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Answer 2

We know that, A−B={x∈A:x/∈B}

Since A⊂B. Therefore, there is no element in A which does not belong to B.

∴A−B=ϕ

Hence, (i) ⇒ (ii).

(ii) ⇒ (ii)

We have, A−B=ϕ⇒A⊂B⇒A∪B=B

Hence, (ii) ⇒ (iii).

(iii) ⇒ (iv)

We have, A∪B=B⇒A⊂B⇒A∩B=A

Hence, (iii) ⇒ (iv).

We have, A∩B=A⇒A⊂B

Hence, (iv)⇒ (ii)

Consequently, (ii) ⇔ (ii) ⇔ (iii) ⇔ (iv).

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