Q12 of 58 Page 12

For all sets A, B and C, show that (A – B) ∩ (A – C) = A – (B ∪ C)

Given: There are three sets A, B and C


To prove: (A – B) ∩ (A – C) = A – (B ∪ C)


Let x ∈ (A – B) ∩ (A – C)


⇒ x ∈ (A – B) and x ∈ (A – C)


⇒ (x ∈ A and x ∉ B) and (x ∈ A and x ∉ C)


⇒ x ∈ A and (x ∉ B and x ∉ C)


⇒ x ∈ A and x ∉ (B ∪ C)


⇒ x ∈ A – (B ∪ C)


⇒ (A – B) ∩ (A – C) ⊂ A – (B ∪ C)………(i)


Let y ∈ A – (B ∪ C)


⇒ y ∈ A and y ∉ (B ∪ C)


⇒ y ∈ A and (y ∉ B and y ∉ C)


⇒ (y ∈ A and y ∉ B) and (y ∈ A and y ∉ C)


⇒ y ∈ (A – B) and y ∈ (A – C)


⇒ y ∈ (A – B) ∩ (A – C)


⇒ A – (B ∪ C) ⊂ (A – B) ∩ (A – C) ………(ii)


We know:


P ⊂ Q and Q ⊂ P ⇒ P = Q


From (i) and (ii):


A – (B ∪ C) = (A – B) ∩ (A – C)


More from this chapter

All 58 →
10

A, B and C are subsets of Universal Set U. If A = {2, 4, 6, 8, 12, 20}, B = {3, 6, 9, 12, 15}, C = {5, 10, 15, 20} and U is the set of all whole numbers, draw a Venn diagram showing the relation of U, A, B and C.

 11

Let U be the set of all boys and girls in a school, G be the set of all girls in the school, B be the set of all boys in the school, and S be the set of all students in the school who take swimming. Some, but not all, students in the school take swimming. Draw a Venn diagram showing one of the possible interrelationship among sets U, G, B and S.

 13

Determine whether each of the statement in

For all sets A and B, (A – B) ∪ (A ∩ B) = A

 14

Determine whether each of the statement in

For all sets A, B and C, A – (B – C) = (A – B) – C