Question

How do I prove that:
A⊆B⊆C ⇔ A∪B=B∩C

Answer 1

Answer:

given A⊆B⊆C

we have to prove that A∪B=B∩C

so

if A⊆B then A∪B=B

if B⊆C then B∩C=B

so if A⊆B⊆C then A∪B=B=B∩C

conversly

given A∪B=B∩C

we have to prove that A⊆B⊆C

so it can be proved by contradiction method

let B⊆A and C⊆B

then A∪B=A and B∩C=C

which contradicts the fact that A∪B=B∩C

so our assumption that B⊆A and C⊆B is wrong

so A⊆B and B⊆C

hence A⊆B⊆C ⇔ A∪B=B∩C

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