Question

Let A, B and C be sets. Then show that

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).​

Answer 1

Answer:

To show that these two sets are equal, we must show that every element in one set is also in the other...

Consider an element x ∈ A ∪ (B ∩ C).

Then either x ∈ A  or else both x ∈ B and x ∈ C.

If x ∈ A, then certainly x ∈ A ∪ B, and also x ∈ A ∪ C.  So x ∈ (A ∪ B) ∩ (A ∪ C).

If on the other hand, we have x ∈ B and x ∈ C, then again x ∈ A ∪ B and x ∈ A ∪ C.  So x ∈ (A ∪ B) ∩ (A ∪ C).

This establishes that A ∪ (B ∩ C) ⊆ (A ∪ B) ∩ (A ∪ C).  That is, every element in the left hand side is also in the right hand side.  Now we go the other way...

Consider an element x ∈ (A ∪ B) ∩ (A ∪ C).​

Then x ∈ A ∪ B and x ∈ A ∪ C.

If x ∈ A, then certainly x ∈ A ∪ (B ∩ C).

Consider then the situation where x ∉ A.

Then since x ∈ A ∪ B, it follows that x ∈ B.

Similarly, since x ∈ A ∪ C, it follows that x ∈ C.

Therefore, x ∈ B ∩ C and so x ∈ A ∪ (B ∩ C).

This establishes that A ∪ (B ∩ C) ⊇ (A ∪ B) ∩ (A ∪ C).  That is, every element in the right hand side is also in the left hand side.

Putting the two together gives the required result.

Hope this helps.

Answer 2

Answer:

hey guys here is your answer

hope it helps