Question

Let A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. show that B = C.​

Answer 1

$\huge\underline\mathfrak\pink{♡Answer♡}$

According to the question,

A ∪ B = A ∪ C

And,

A ∩ B = A ∩ C

To show,

B = C

Let us assume,

x ∈ B

So,

x ∈ A ∪ B

x ∈ A ∪ C

Hence,

x ∈ A or x ∈ C

When x ∈ A, then,

x ∈ B

∴ x ∈ A ∩ B

As, A ∩ B = A ∩ C

So, x ∈ A ∩ C

∴ x ∈ A or x ∈ C

x ∈ C

∴ B ⊂ C

Similarly,

it can be shown that C ⊂ B

Hence, B = C

Answer 2

Answer:

➡According to the question,

A ∪ B = A ∪ C

And,

A ∩ B = A ∩ C

➡To show,

B = C

➡Let us assume,

x ∈ B

So,

x ∈ A ∪ B

x ∈ A ∪ C

➡Hence,

x ∈ A or x ∈ C

When x ∈ A, then,

x ∈ B

∴ x ∈ A ∩ B

As, A ∩ B = A ∩ C

So, x ∈ A ∩ C

∴ x ∈ A or x ∈ C

x ∈ C

∴ B ⊂ C

Similarly,

it can be shown that C ⊂ B

➡Hence, B = C

Step-by-step explanation:

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