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

Write the following as intervals: $\{x : x \in \mathbb{R}, 0 \le x < 7\}$

Write down all the subsets of the following set: $\{1, 2, 3\}$.

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

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 $A \subset B$ and $B \subset C$,then $A \subset C$.

If $aN = \{ ax : x \in N \}$ and $bN \cap cN = dN$,where $b, c \in N$ are relatively prime,then