Question

Prove that,

a intersection [b-c] = [a intersection b] - [a intersection c]

Answer 1

Answer:

Step-by-step explanation:

let

    x ∈ A∩(B-C)

⇒  x ∈ A and x ∈ (B-C)

From x ∈ (B-C)

⇒  x ∈ B and x ∉ C

NOW

Since

    x ∈ A and x ∈ B

So

    x ∈ (A∩B)......(1)

And Also

    x ∈ A and x ∉ C

⇒  x ∉ (A∩C)  ....(2)

From (1) and (2) it is clear

     x ∈ (A∩B) - (A∩C)

Thus

A∩(B-C) ⊂ (A∩B) - (A∩C)    .......(3)

NOW

let

     y ∈ (A∩B) - (A∩C)

Then

⇒   y ∈ (A∩B)  and y ∉ (A∩C)

Since  y ∈ A and y ∈ B and also y ∉  (A∩C)

This implies

       y ∉ C

And

Since y ∈ B and y ∉ C this implies

       y ∈ (B-C)

And also

       y ∈ A  and y ∈  (B-C)  

so

       y ∈ A∩(B-C)

Thus

(A∩B) - (A∩C) ⊂ A∩(B-C)   .....(4)

From (3) and (4)

A∩(B-C) = (A∩B) - (A∩C