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

Question:

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) False

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

Now,  $2 \in\{1,2\}$ and $\{1,2\} \in\{\{3\}, 1,\{1,2\}\}$

$\therefore A \in B$

However, $2 \notin\{\{3\}, 1,\{1,2\}\}$

(ii) False

Let $\mathrm{A}=\{2\}, \mathrm{B}=\{0,2\}$, and $\mathrm{C}=\{1,\{0,2\}, 3\}$

As A ⊂ B

B ∈ C

However, $\mathrm{A} \notin \mathrm{C}$

(iii) True

Let A ⊂ B and B ⊂ C.

Let x ∈ A

$\Rightarrow x \in \mathrm{B} \quad[\because \mathrm{A} \subset \mathrm{B}]$

$\Rightarrow x \in \mathrm{C} \quad[\because \mathrm{B} \subset \mathrm{C}]$

∴ A ⊂ C

(iv) False

Let $\mathrm{A}=\{1,2\}, \mathrm{B}=\{0,6,8\}$, and $\mathrm{C}=\{0,1,2,6,9\}$

Accordingly, $\mathrm{A} \nsubseteq \mathrm{B}$ and $\mathrm{B} \not \mathrm{C}$.

However, $A \subset C$

(v) False

Let A = {3, 5, 7} and B = {3, 4, 6}

Now, 5 ∈ A and A ⊄ B

However, 5 ∉ B

(vi) True

Let A ⊂ B and ∉ B.

To show: x ∉ A

If possible, suppose x ∈ A.

Then, x ∈ B, which is a contradiction as ∉ B

x ∉ A