Question

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

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).
Do not show in words.
Show by calculating (I'll mark you brainliest) ​

Answer 1

Answer:

Since B ∩ C ⊆ B and B ∩ C ⊆ C, we have

C, we haveA ∪ (B∩C) ⊆ A ∪ B

Band

BandA ∪ (B∩C) ⊆ A ∪ C

CThis shows that A ∪ (B∩C) is contained in both A∪B and A∪C, so it is contained in their intersection:

(B∩C) is contained in both A∪B and A∪C, so it is contained in their intersection:A ∪ (B∩C) ⊆ (A∪B)∩(A∪C)

(A∪B)∩(A∪C)This proves containment in one direction.

(A∪B)∩(A∪C)This proves containment in one direction.For the opposite direction, suppose that x∈(A∪B) ∩ (A∪C). There are two possibilities: either x∈A or x∉A.

(A∪C). There are two possibilities: either x∈A or x∉A.If x∈A then certainly x ∈ A∪(B∩C).

A∪(B∩C).On the other hand, if x∉A, then x must be in both B and C, since x∈(A∪B)∩(A∪C). Consequently, x∈B∩C, and therefore x∈A∪(B∩C).

A∪(B∩C).In both cases we have x∈A∪(B∩C). This proves the containment

A∪(B∩C).In both cases we have x∈A∪(B∩C). This proves the containment(A∪B)∩(A∪C)⊆A∪(B∩C)