**Question:**

if A = {3, {4, 5}, 6} find which of the following statements are true.

(i) $\{4,5\} \nsubseteq \mathrm{A}$

(ii) $\{4,5\} \in A$

(iii) $\{\{4,5\}\} \subseteq A$

(iv) $4 \in A$

(v) $\{3\} \subseteq A$

(vi) $\{\phi\} \subseteq A$

(vii) $\phi \subseteq A$

(viii) $\{3,4,5\} \subseteq A$

(ix) $\{3,6\} \subseteq A$

**Solution:**

(i) True

Explanation: we have, $A=\{3,\{4,5\}, 6\}$

Let $\{4,5\}=x$

Now, $A=\{3, x, 6\}$

4,5 is not in $A,\{4,5\}$ is an element of $A$ and element cannot be subset of set,thus $\{4,5\}$ $\not \subset \mathrm{A}$.

(ii) True

Explanation: we have, $A=\{3,\{4,5\}, 6\}$

Let $\{4,5\}=x$

Now, $A=\{3, x, 6\}$

Now, $x$ is in $A$.

So, $x \in A$.

Thus, $\{4,5\} \in \mathrm{A}$

(iii) True

Explanation: $\{4,5\}$ is an element of set $\{\{4,5\}\}$.

Let $\{4,5\}=x$

$\{\{4,5\}\}=\{x\}$

we have, $A=\{3,\{4,5\}, 6\}$

Now, $A=\{3, x, 6\}$

So, $x$ is in $\{x\}$ and $x$ is also in $A$.

So, $\{x\}$ is a subset of $A$.

Hence, $\{\{4,5\}\} \subseteq \mathrm{A}$

(iv) False

Explanation: 4 is not an element of A.

(v) True

Explanation: 3 is in {3} and also 3 is in A.

(vi) False

Explanation: $\phi$ is an element in $\{\phi\}$ but not in $A$.

Thus, $\{\phi\} \not \subset \mathrm{A}$

(vii) True

Explanation: $\phi$ is a subset of every set.

(viii) False

Explanation: we have, $A=\{3,\{4,5\}, 6\}$

Let $\{4,5\}=x$

Now, $A=\{3, x, 6\}$

4,5 is in $\{3,4,5\}$ but not in $A$, thus $\{3,4,5\} \not \subset A$.

(ix) True

Explanation: 3,6 is in $\{3,6\}$ and also in $\mathrm{A}$, thus $\{3,6\} \subseteq \mathrm{A}$.