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 $S = \{ 0,\,1,\,5,\,4,\,7\} $. Then the total number of subsets of $S$ is

Find the pairs of equal sets, if any, give reasons:

$A = \{ 0\} ,$

$B = \{ x:x\, > \,15$ and $x\, < \,5\} $

$C = \{ x:x - 5 = 0\} ,$

$D = \left\{ {x:{x^2} = 25} \right\}$

$E = \{ \,x:x$ is an integral positive root of the equation ${x^2} - 2x - 15 = 0\,\} $

For an integer $n$ let $S_n=\{n+1, n+2, \ldots \ldots, n+18\}$. Which of the following is true for all $n \geq 10$ ?

Write the following as intervals :

$\{ x:x \in R, - 12\, < \,x\, < \, - 10\} $

State which of the following sets are finite or infinite :

$\{ x:x \in N$ and $(x - 1)(x - 2) = 0\} $