Question

3.
Let A, B, and C be the sets such that AUB=AUC and A intersection B = A intersection C. Show
that B = C.
​

Answer 1

Answer:Let x ∈ B

There can be two cases, x ∈ A or x ∉ A.

Case -1 x ∈ A

As x ∈ A and x ∈ B so that x ∈ ( A ∩ B)

Given that A∩ B = A ∩ C, so that

x ∈ ( A ∩ C)

x ∈ A and x ∈ C

x ∈ B then x ∈ C so that B ⊂ C.

Case -2, x ∉ A

We have already assumed that x ∈ B.

Hence, x ∈ (A ∪ B)

Given that A ∪ B = A ∪ C, so that

⇒ x ∈ (A ∪ C)

⇒x ∈ A or x ∈ C

But we assumed that x ∉ A Hence x ∈ C

x ∈ B then x ∈ C so that B ⊂ C.

So in both cases B ⊂ C. ...(1)

Similarly, we can prove C ⊂ B ...(2)

From equation (1) and (2)

B= C

Step-by-step explanation:

Answer 2

Answer:

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Step-by-step explanation:

Step-1:Prove using suitable formula of sets.

It is given that,

A∪B = A∪C …(1)

A∩B = A∩C…(2)

Taking  ’∩ C’ on both sides in equation (1)

(A∪B)∩C = (A∪C)∩C

We know that,

(A∪B)∩C = (A∩C)∪(B∩C) and (A∪C)∩C = C

So,

(A∩C)∪(B∩C)=C

(A∩B)∪(B∩C)=C…(3)[From(2))

Again,

Taking  ’∩ B’ on both side in equation (1)

(A∪B)∩B = (A∪C)∩B

B = (A∩B)∪(C∩B)

B = (A∩B)∪(B∩C)

B = C[From (3)]

Hence, proved.