In each of the following, determine whether ...

NCERT Class XI Mathematics - Sets - Solutions

Question : 59 of 73

Marks: +1, -0

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

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.

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