NCERT Class XI Mathematics - Sets - Solutions
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Question : 59
Total: 73
In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.
(i) If x ∈ A and A ∈ B, then x ∈ B
(ii) If A ⊂ B and B ∈ C, then A ∈ C
(iii) If A ⊂ B and B ⊂ C, then A ⊂ C
(iv) If A ⊄ B and B ⊄ C, then A ⊄ C
(v) If x ∈ A and A ⊄ B, then x ∈ B
(vi) If A ⊂ B and x ∉ B, then x ∉ A
(i) If x ∈ A and A ∈ B, then x ∈ B
(ii) If A ⊂ B and B ∈ C, then A ∈ C
(iii) If A ⊂ B and B ⊂ C, then A ⊂ C
(iv) If A ⊄ B and B ⊄ C, then A ⊄ C
(v) If x ∈ A and A ⊄ B, then x ∈ B
(vi) If A ⊂ B and x ∉ B, then x ∉ A
Solution:
(i) The statement is false.
Take A = {1}, B = {{1}, 2} Now 1 ∈ A and A ∈ B, but 1 ∉ B.
(ii) The statement is false.
Take A = {1}, B = {1, 2}, C = {{1, 2}, 3}
Now A ⊂ B and B ∈ C but A ∉ C.
(iii) The statement is true.
Let x ∈ A ⇒ x ∈ B (Since A ⊂ B)
⇒ x ∈ C (Since B ⊂ C)
Now x ∈ A ⇒ x ∈ C ∴ A ⊂ C.
(iv) The statement is false.
Take A = {1, 2} B = {2, 3}, C = {1, 2, 5}
Now A ⊄ B and B ⊄ C, but A ⊂ C.
(v) The statement is false.
Take A = {1, 2} and B = {2, 3, 4, 5}
Now 1 ∈ A and A ⊄ B but 1 ∉ B.
(vi) The statement is true.
Let x ∈ A ⇒ x ∈ B (Since A ⊂ B)
Now x ∉ B ⇒ x ∉ A.
Take A = {1}, B = {{1}, 2} Now 1 ∈ A and A ∈ B, but 1 ∉ B.
(ii) The statement is false.
Take A = {1}, B = {1, 2}, C = {{1, 2}, 3}
Now A ⊂ B and B ∈ C but A ∉ C.
(iii) The statement is true.
Let x ∈ A ⇒ x ∈ B (Since A ⊂ B)
⇒ x ∈ C (Since B ⊂ C)
Now x ∈ A ⇒ x ∈ C ∴ A ⊂ C.
(iv) The statement is false.
Take A = {1, 2} B = {2, 3}, C = {1, 2, 5}
Now A ⊄ B and B ⊄ C, but A ⊂ C.
(v) The statement is false.
Take A = {1, 2} and B = {2, 3, 4, 5}
Now 1 ∈ A and A ⊄ B but 1 ∉ B.
(vi) The statement is true.
Let x ∈ A ⇒ x ∈ B (Since A ⊂ B)
Now x ∉ B ⇒ x ∉ A.
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