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:

In each of the following, let us determine whether the statement is true or false.

(i) If x ∈ A and A ∈ B, then x ∈ B

It is FALSE as the notation A ∈ B is incorrect.

Placing ∈ symbol between two sets is incorrect.

(ii) If A ⊂ B and B ∈ C, then A ∈ C

It is FALSE as the notations B ∈ C, then A ∈ C are incorrect.

Placing ∈ symbol between two sets is incorrect.

(iii) If A ⊂ B and B ⊂ C, then A ⊂ C

It is given that; A ⊂ B and B ⊂ C

Let x ∈ A

Now,

x ∈ B [Since  A ⊂ B]

x ∈ C [Since  B ⊂ C]

Hence, A ⊂ C.

Thus, the given statement is TRUE.

(iv) If A ⊄ B and B ⊄ C , then A ⊄ C

Let A = {1, 2}, B = {3, 4} and C = {0, 1, 2, 5}

Here, {1, 2} ⊄ {3, 4} and {3, 4} ⊄ {0, 1, 2, 5}

However, {1, 2} ⊂ {0,1, 2, 5}

i.e., A ⊂ C

Thus, the given statement is FALSE.

(v) If x ∈ A and A ⊄ B, then x ∈ B

Let A = {1, 2, 3} and B = {3, 4, 5}

Let x = 2.

Here, 2 ∈ {1, 2, 3} and {1, 2, 3} ⊄ {3, 4, 5}

However, 2 ∉ B

i.e., x ∉ B

Thus, the given statement is FALSE.

(vi) If A ⊂ B and x ∉ B, then x ∉ A

Let x ∈ A, if possible.

Then x ∈ B [∵ A ⊂ B]

But it is given that x ∉ B.

Which is a contradiction

So, x ∉ A

Thus, the given statement is TRUE

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NCERT Solutions Class 11 Maths Chapter 1 Exercise ME Question 2

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

Summary:

We are asked to determine whether the statement is true or false in each of the following. We found that
(i) If x ∈ A and A ∈ B, then x ∈ B: FALSE
(ii) If A ⊂ B and B ∈ C, then A ∈ C: FALSE
(iii) If A ⊂ B and B ⊂ C, then A ⊂ C: TRUE
(iv) If A ⊄ B and B ⊄ C , then A ⊄ C: FALSE
(v) If x ∈ A and A ⊄ B , then x ∈ B: FALSE
(vi) If A ⊂ B and x ∉ B , then x ∉ A: TRUE