(a)

\begin{align*} A & = \{ 1, 3, 5, 7, 9 \} \end{align*}

(b)(i)

$$ \text{True} $$

(b)(ii)

$$ \text{True} $$

(b)(iii)

$$ \text{False} $$

(b)(iv)

$$ \text{False} $$

(c)(i)

$$ -3 \notin A $$

(c)(ii)

$$ 3 \in A $$

(c)(iii)

$$ 0 \notin A $$

(c)(iv)

$$ 9 \in A $$

(a)

$$ B = \{ 2 \} $$ $$ B \ne \emptyset $$

(b)

$$ C = \{ \text{Saturday, Sunday} \} $$ $$ C \ne \emptyset $$

(c) Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 and Multiples of 9: 9, 18, 27, ...

$$ D = \emptyset $$

(d) A quadrailteral has 4 sides and the sum of interior angles is 360°. The quadrilateral with angles 100°, 100° 100° and 60° have three obtuse angles.

$$ E \ne \emptyset $$

(a)

$$ A' = \{ -4, -2, 0, 1, 3 \} $$

(b)

$$ B' = \emptyset $$

(c) A prime number is a positive integer with only 2 factors, 1 and itself

$$ C = \{ 2, 3 \} $$ $$ C' = \{ -5, -4, -3, -2, -1, 0, 1 \} $$

(d)

$$ D = \{ 3 \} $$ $$ D' = \{ -5, -4, -3, -2, -1, 0, 1, 2 \} $$

(a)

$$ \text{True, since } X \text{ is a proper subset of } Y, \text{ every element in } X \text{ is an element in } Y $$

(b)

$$ \text{False, since } b \text{ is in set } Y \text{ but may not be in the subset } X $$

(c)

$$ \text{True} $$

(d)

$$ \text{True, since } X' \text{ is an empty set, then } X \text{ contains all the elements in the universal set} $$

(i)

\begin{align*} \xi & = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23 \} \\ \\ A & = \{ 4, 8, 12, 16, 20 \} \\ \\ B & = \{ 1, 2, 3, 4, 6, 9, 12, 18 \} \end{align*}

(ii)

\begin{align*} A \cup B' & = \{ 4, 5, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23 \} \end{align*}

(i)

\begin{align*} \xi & = \{ -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7 \} \\ \\ A & = \{ -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 \} \\ \\ B & = \{ 1, 2, 3, 4, 5, 6, 7 \} \\ \\ A' & = \{ -7, 7 \} \end{align*}

(ii)

\begin{align*} A \cap B & = \{ 1, 2, 3, 4, 5, 6 \} \end{align*}

(iii)

\begin{align*} B' & = \{ -7, -6, -5, -4, -3, -2, -1, 0 \} \\ \\ A \cup B' & = \{ -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 \} \end{align*}

(i)

\begin{align*} \xi & = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 \} \\ \\ A & = \{ 1^2, 2^2, 3^2 \} \\ & = \{ 1, 4, 9 \} \\ \\ B & = \{ 1, 2, 13 \} \end{align*}

(ii)

\begin{align*} A' \cap B' & = \{ 3, 5, 6, 7, 8, 10, 11, 12, 14, 15 \} \end{align*}

(i)

\begin{align*} \xi & = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 \} \\ \\ A & = \{ 5, 10, 15, 20 \} \\ \\ B & = \{ 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20 \} \end{align*}

(ii)

\begin{align*} A \cap B' & = \{ 5 \} \end{align*}

$$ \{ \}, \{ \text{s} \}, \{ \text{i} \}, \{ \text{t} \}, \{ \text{s, i} \}, \{ \text{s, t} \}, \{ \text{i, t} \} $$

A rational number is a number that can be expressed as a fraction. An integer is a number without any decimal or fractional component.