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

Examine whether the following statement is true or false:
$\{a\} \subset \{a, b, c\}$

Two dice are thrown. The events $A$,$B$,and $C$ are as follows:
$A$: getting an even number on the first die.
$B$: getting an odd number on the first die.
$C$: getting the sum of the numbers on the dice $\leq 5$.
State true or false: (give reason for your answer)
Statement: $A$ and $B^{\prime}$ are mutually exclusive.

In the set-builder notation (rule method),the null set is represented by:

Write the set $\{ x : x \text{ is a positive integer and } x^2 < 40 \}$ in the roster form.

State whether the following set is finite or infinite:
The set of animals living on the earth.