NCERT Class XI Mathematics - Sets - Solutions
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Question : 61
Total: 73
Show that the following four conditions are equivalent :
(i) A ⊂ B (ii) A – B = f (iii) A ∪ B = B (iv) A ∩ B = A
(i) A ⊂ B (ii) A – B = f (iii) A ∪ B = B (iv) A ∩ B = A
Solution:
(i) ⇒ (ii)
A – B = {x : x ∈ A and x ∉ B}
Since A ⊂ B
∴ A – B = f
(ii) ⇒ (iii)
A – B = f ⇒ A ⊂ B ⇒ A ∪ B = B
(iii) ⇒ (iv)
A ∪ B = B ⇒ A ⊂ B ⇒ A ∩ B = A
(iv) ⇒ (i)
A ∩ B = A ⇒ A ⊂ B
Thus (i) ⇔ (ii) ⇔ (iii) ⇔ (iv).
A – B = {x : x ∈ A and x ∉ B}
Since A ⊂ B
∴ A – B = f
(ii) ⇒ (iii)
A – B = f ⇒ A ⊂ B ⇒ A ∪ B = B
(iii) ⇒ (iv)
A ∪ B = B ⇒ A ⊂ B ⇒ A ∩ B = A
(iv) ⇒ (i)
A ∩ B = A ⇒ A ⊂ B
Thus (i) ⇔ (ii) ⇔ (iii) ⇔ (iv).
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