## For two sets A and B , prove that A intersection B ^ c = phy that implies a is a subset of b​

Question

For two sets A and B , prove that A intersection B ^ c = phy that implies a is a subset of b​

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4 weeks 2021-11-02T06:54:27+00:00 1 Answer 0 views 0

First, assume that A ⊆B and we have to prove that A ∩B = A

Proof: Given A ⊆B

Let x ∈ A ∩ B which implies that x ∈ A and x ∈B. Hence we can imply that

A ∩ B ⊆ A (We started with an element in A ∩ B and concluded that it is in A)

——(1)

Now, let x ∈ A which implies x ∈B (as A ⊆B)

Hence, x ∈ A ∩ B, which in turn implies

A ⊆ A ∩ B ——-(2)

From (1) and (2), A ∩B = A

Conversely, assume A ∩B = A and we have to prove that A ⊆B

Proof: Given A ∩B = A

Let x ∈ A , then since A ∩B = A , x ∈ A ∩ B

This implies x ∈B, hence in turn implies A ⊆B (We have an element in A and we proved that it is in B).