Q23 of 58 Page 12

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

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Given: A, B and C are three given sets


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


Let x ∈ A ∩ (B ∪ C)


⇒ x ∈ A and x ∈ (B ∪ C)


⇒ x ∈ A and (x ∈ B or x ∈ C)


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


⇒ x ∈ A ∩ B or x ∈ A ∩ C


⇒ x ∈ (A ∩ B) ∪ ( A ∩ C)


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


Let y ∈ (A ∩ B) ∪ (A ∩ C)


⇒ y ∈ A ∩ B or x ∈ A ∩ C


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


⇒ y ∈ A and (y ∈ B or y ∈ C)


⇒ y ∈ A and y ∈ (B ∪ C)


⇒ y ∈ A ∩ (B ∪ C)


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


We know:


P ⊂ Q and Q ⊂ P ⇒ P = Q


From (i) and (ii):


A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)


Hence Proved


More from this chapter

All 58 →
21

Using properties of sets prove the statements given

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

 22

Using properties of sets prove the statements given

Let . Is T an empty set? Justify your answer.

 24

Out of 100 students; 15 passed in English, 12 passed in Mathematics, 8 in Science, 6 in English and Mathematics, 7 in Mathematics and Science; 4 in English and Science; 4 in all the three. Find how many passed

(i) in English and Mathematics but not in Science


(ii) in Mathematics and Science but not in English


(iii) in Mathematics only


(iv) in more than one subject only

 25

In a class of 60 students, 25 students play cricket and 20 students play tennis, and 10 students play both the games. Find the number of students who play neither?