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

What is the remainder when the polynomial $P(x) = 1 + x + x^3 + x^9 + x^{27} + x^{81} + x^{243}$ is divided by $x - 1$?

Show that $A \cap B = A \cap C$ does not necessarily imply $B = C$.

Write the following interval in set-builder form: $\left[ -23, 5 \right)$

Make correct statements by filling in the symbols $\subset$ or $\not\subset$ in the blank spaces:
$\{a, b, c\} \dots \{b, c, d\}$

State whether the following statement is true or false. Justify your answer.
${2, 6, 10}$ and ${3, 7, 11}$ are disjoint sets.