Show that the following four conditions are equivalent : (i) A ⊂ B (ii) A – B = φ (iii) A ∪ B = B (iv) A ∩ B = A

Solution:

To prove (i) is equivalent to (ii):
From (i) A ⊂ B
It means that all elements of A are in B.
i.e., there is no element of A that is NOT in B.
i.e., A – B = Φ (which is (ii))
Thus, (i) ⇔ (ii)
 

To prove (i) is equivalent to (iii):
From (i) A ⊂ B.
It means that all elements of A are in B.
Then A ∪ B = Set of all elements of A and B =  B (which is (iii))
Thus, (i) ⇔ (iii)
 

To prove (i) is equivalent to (iv):
From (i) A ⊂ B.
It means that all elements of A are in B.
Then A ∩ B = Set of all elements common to A and B = B (which is (iv))
Thus, (i) ⇔ (iv)

We have proved that  (i) ⇔ (ii),  (i) ⇔ (iii), and  (i) ⇔ (iv).

Thus,

(i) ⇔ (ii) ⇔ (iii) ⇔ (iv)

NCERT Solutions Class 11 Maths Chapter 1 Exercise ME Question 4

Show that the following four conditions are equivalent : (i) A ⊂ B(ii) A – B = φ (iii) A ∪ B = B (iv) A ∩ B = A

Summary:

We are asked to prove the given four conditions are equivalent. We have proved that A ⊂ B ⇔ A – B = Φ ⇔ A υ B = B ⇔ A ∩ B = A.