State which of the following sets are finite or infinite: $\{ x:x \in N \text{ and } 2x - 1 = 0 \}$

Prove that for any sets $A$ and $B$,$A \cap (A \cup B) = A$.

In each of the following,determine whether the statement is true or false. If it is true,prove it. If it is false,give a counterexample.
If $x \in A$ and $A \not\subset B$,then $x \in B$

Find the pairs of equal sets,if any,give reasons:
$A = \{ 0 \}$
$B = \{ x : x > 15 \text{ and } x < 5 \}$
$C = \{ x : x - 5 = 0 \}$
$D = \{ x : x^2 = 25 \}$
$E = \{ x : x \text{ is an integral positive root of the equation } x^2 - 2x - 15 = 0 \}$

The remainder obtained when the polynomial $1 + x + x^3 + x^9 + x^{27} + x^{81} + x^{243}$ is divided by $x - 1$ is

State which of the following sets are finite or infinite:
$\{ x:x \in N \text{ and } x^2 = 4 \}$