Q6 of 58 Page 12

If A and B are subsets of the universal set U, then show that

(i) A ⊂ A ∪ B


(ii) A ⊂ B ⇔ A ∪ B = B


(iii) (A ∩ B) ⊂ A

(i) Given: A and B are two subsets


To prove: A ⊂ A ∪ B


Let x ∈ A


⇒ x ∈ A or x ∈ B


⇒ x ∈ A ∪ B


⇒ A ⊂ A ∪ B


Hence Proved


(ii) Given: A and B are two sets


To prove: A ⊂ B ⇔ A ∪ B = B


Let x ∈ A ∪ B


⇒ x ∈ A or x ∈ B


⇒ x ∈ B {∵ A ⊂ B}


⇒ A ∪ B ⊂ B…………(1)


We know,


B ⊂ A ∪ B {this is always true}…………(2)


From (1) and (2):


A ∪ B = B


Now,


Let y ∈ A


⇒ y ∈ A ∪ B


⇒ y ∈ B {∵ A ∪ B = B}


⇒ A ⊂ B


So,


A ⊂ B ⇔ A ∪ B = B


Hence Proved


(iii) Given: A and B are two subsets


To prove: (A ∩ B) ⊂ A


Let x ∈ A ∩ B


⇒ x ∈ A and x ∈ B


⇒ x ∈ A


⇒ A ∩ B ⊂ A


Hence Proved


More from this chapter

All 58 →
4

State which of the following statements are true and which are false. Justify your answer.

(i) 35 ∈ {x | x has exactly four positive factors}.


(ii) 128 ∈ {y | the sum of all the positive factors of y is 2y}


(iii) 3 ∉ {x | x4 – 5x3 + 2x2 – 112x + 6 = 0}


(iv)496 ∉ {y | the sum of all the positive factors of y is 2y}.

 5

Given L = {1, 2, 3, 4}, M = {3, 4, 5, 6} and N = {1, 3, 5}

Verify that L – (M ∪ N) = (L – M) ∩ (L – N)

 7

Given that N = {1, 2, 3 ,..., 100}. Then write

(i) the subset of N whose elements are even numbers.


(ii) the subset of N whose element are perfect square numbers.

 8

If X = {1, 2, 3}, if n represents any member of X, write the following sets containing all numbers represented by

(i) 4n


(ii) n + 6


(iii)


(iv) n – 1