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

Question

Philoid · Accepted Answer

(i) Let us assume A = {1, 2}

And, B = {1, {1, 2}, {3}}

So,

And,

∴

But,

Hence, the given statement is false

(ii) Let us assume, A {2}

B = {0, 2}

And, C = {1, {0, 2}, 3}

It is given in the question that,

∴

But,

Hence, the given statement is false

(iii) It is given in the question that,

Let us assume,

So,

And,

∴

Hence, the given statement is correct

(iv) It is given in the question that,

A ⊄ B

And, B ⊄ C

Let us now assume, A = {1, 2}

B = {0, 6, 8}

And, C = {0, 1, 2, 6, 9}

∴

Hence, the given statement is false

(v) It is given in the question that,

And, A ⊄ B

Let us now assume, A = {3, 5, 7}

And, B = {3, 4, 6}

As,

And,

∴

Hence, the given statement is false

(vi) It is given in the question that,

A ⊂ B

And, x ∉ B

Let us suppose, then we have:

But it is given in the question that, x ∉ B

∴ x ∉ A

Hence, the given statement is true