In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false

(i) Right answer is True

Explanation:-

x∈A and A∈B, then x∈B

To determine whether the statement is true

A = {1, 2} and B = {1, {1, 2}, {3}}

Now, we have,

2 ∈ {1, 2} and {1, 2} ∈ {1, {1, 2}, {3}}

Hence, we get,

A ∈ B

We also know,

{2} ∉ {1, {1, 2}, {3}}

(ii)  Right answer is False

Explanation:-

Let us assume that,

A {2}

B = {0, 2}

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

From the question,

A ⊂ B

Hence,

B ∈ C

But, we know,

A ∉ C

(iii)  Right answer is True

Explanation:-

A ⊂ B and B ⊂ C

Let us assume that,

x ∈ A

Then, we have,

x ∈ B

And,

x ∈ C

Therefore,

A ⊂ C

(iv) Right answer is False

Explanation:-

A ⊄ B

Also,

B ⊄ C

Let us assume that,

A = {1, 2}

B = {0, 6, 8}

And,

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

A ⊂ C

(v) Right answer is False

Explanation:-

x ∈ A

Also,

A ⊄ B

Let us assume that,

A = {3, 5, 7}

Also,

B = {3, 4, 6}

We know that,

A ⊄ B

5 ∉ B

(vi) Right answer is True

Explanation:-

A ⊂ B 

Also,

x ∉ B

Let us assume that,

x ∈ A,

We have,

x ∈ B,

From the question,

We have, x ∉ B

x ∉ A