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.

If $x \in A$ and $A \in B,$ then $x \in B$

Let $A=\{1,2,\{3,4\}, 5\} .$ Which of the following statements are incorrect and why ?

$\{1,2,5\}\subset A$

Which of the following pairs of sets are equal ? Justify your answer.

$\mathrm{X} ,$ the set of letters in $“\mathrm{ALLOY}"$ and $\mathrm{B} ,$ the set of letters in $“\mathrm{LOYAL}”.$

Let $A=\{1,2,3,4,5,6\} .$ Insert the appropriate symbol $\in$ or $\notin$ in the blank spaces:

$ 8\, .......\, A $

Set $A$ has $m$ elements and Set $B$ has $n$ elements. If the total number of subsets of $A$ is $112$ more than the total number of subsets of $B$, then the value of $m \times n$ is

Write the following sets in the set-builder form :

$\{ 3,6,9,12\}$