(a)

\begin{align} A & = \{ \text{m, a, t, h, e, i, c, s} \} \\ \\ n(A) & = 8 \end{align}

(b)

\begin{align} B & = \{ \text{red, white} \} \\ \\ n(B) & = 2 \end{align}

(c)

\begin{align} C & = \{ \text{o} \} \\ \\ n(C) & = 1 \end{align}

(d)

\begin{align} D & = \{ \text{} \} \\ \\ n(D) & = 0 \end{align}

(a)

\begin{align} P & = \{ 5, 0, 1, 3, 4 \} \\ \\ P & \text{ is a finite set} \end{align}

(b)

\begin{align} 2x - 1 & < 5 \\ 2x & < 5 + 1 \\ 2x & < 6 \\ x & < {6 \over 2} \\ x & < 3 \\ \\ Q & = \{ 2, 1, 0, -1, ... \} \\ \\ Q & \text{ is an infinite set} \end{align}

(c)

\begin{align} (x + 2)(x - 5) & = 0 \end{align}
\begin{align} x + 2 & = 0 && \text{ or } & x - 5 & = 0 \\ x & = -2 &&& x & = 5 \end{align}
\begin{align} R & = \{ -2, 5 \} \\ \\ R & \text{ is a finite set} \end{align}

(d)

\begin{align} S & = \{ 0, 1, 2, 3, ... \} \\ \\ S & \text{ is an infinite set} \end{align}

(a)

\begin{align} \{x:x \text{ is a factor of 12} \} \end{align}

(b)

\begin{align} \{x:x \text{ is a multiple of 3} \} \end{align}

(c)

\begin{align} \{x:x \text{ is a prime number} \} \end{align}

(d)

\begin{align} 1^3, 2^3, 3^3, 4^3, ... & = 1, 8, 27, 64, ... \\ \\ \{x:x & \text{ is a perfect cube} \} \end{align}

(a)

$$ \text{True} $$

(b)

$$ \text{False (since } \{12, 16\} \text{ is a set)} $$

(c)

$$ \text{False} $$

(d)

$$ \text{False} $$

(a)

$$ \text{True} $$

(b)

$$ \text{True} $$

(c)

$$ \text{True} $$

(d)

$$ \text{False (since } \{ \text{a} \} \text{ is a set)} $$

(e)

$$ \text{True} $$

(f)

$$ \text{True} $$

(a)

\begin{align} \{ \}, \{ 3 \}, \{ 6 \}, \{ 9 \}, \{ 3, 6 \}, \{3, 9 \}, \{ 6, 9 \} \end{align}

(b)(i)

$$ 1 $$

(b)(ii)

$$ 3 $$

(b)(iii)

$$ 3 $$

(a)

\begin{align} A & = \{ 1, 2, 3 \} \\ \\ (x - 1)(x - 2) & = 0 \end{align}
\begin{align} x - 1 & = 0 && \text{ or } & x - 2 & = 0 \\ x & = 1 &&& x & = 2 \end{align}
\begin{align} B & = \{ 1, 2 \} \\ \\ \therefore B & \subset A \end{align}

(b)

\begin{align} A & = C \end{align}

(c)

\begin{align} B & \subset D \end{align}

(d)

\begin{align} C & \not\subset D \end{align}

(a)

\begin{align} \xi & = \{ 2, 4, 6, 8, 10, 12, 14, ... \} \\ \\ A & = \{ 2, 4, 6, 8, 12, 24 \} \end{align}

(b)

\begin{align} B & = \{ 2, 4, 6, 12, 18, 36 \} \\ \\ n(B) & = 6 \end{align}

(c)

\begin{align} A & = \{ 2, 4, 6, 8, 12, 24 \} \\ \\ B & = \{ 2, 4, 6, 12, 18, 36 \} \\ \\ A & \not \subset B \end{align}

(d)

\begin{align} A & = \{ 2, 4, 6, 8, 12, 24 \} \\ \\ B & = \{ 2, 4, 6, 12, 18, 36 \} \\ \\ B & \not \subset A \end{align}

(a)

\begin{align} 2x + 1 & < 5 \\ 2x & < 5 - 1 \\ 2x & < 4 \\ x & < {4 \over 2} \\ x & < 2 \\ \\ \xi & = \{ -3, -2, -1, 0, 1, 2, 3 \} \\ \\ A & = \{ -3, -2, -1, 0, 1 \} \end{align}

(b)

$$ \text{No since } A = \{ -3, -2, -1, 0, 1 \} $$

(c)

\begin{align} \xi & = \{ -3, -2, -1, 0, 1, 2, 3 \} \\ \\ B & = \{ -3, 0, 3 \} \\ \\ n(B) & = 3 \end{align}

(d)

\begin{align} A & = \{ -3, -2, -1, 0, 1 \} \\ \\ B & = \{ -3, 0, 3 \} \\ \\ \text{Commmon elements: } & -3, 0 \end{align}

(a)(i)

\begin{align} 3^2 + 4^2 & = 25 \\ \\ 5^2 & = 25 \\ \\ \text{By converse of Pythagroas} & \text{ theorem, triangle is right-angled and in } A \end{align}

(a)(ii)

\begin{align} 2^2 + 2^2 & = 8 \\ \\ 3^2 & = 9 \\ \\ \text{Triangle is not} & \text{ right-angled and not in } A \end{align}

(b)(i)

\begin{align} & \text{Yes, since all right-angled triangles are triangles and } A \ne U \end{align}

(b)(ii)

\begin{align} & \text{Yes, since all right-angled triangles are 3-sided polygons and } A \ne V \end{align}

(a)

\begin{align} & \text{False, since } \{ \text{a, b} \} \text{ is a set} \end{align}

(b)

\begin{align} & \text{False, since } A \text{ and } \xi \text{ are sets} \end{align}

(c)

\begin{align} & \text{False, since } A = A \end{align}

(d)

\begin{align} & \text{True} \end{align}

(e)

\begin{align} & \text{False, since } \{ \emptyset \} \text{ is a set with an element } \emptyset \end{align}

(a)(i)

\begin{align} & \text{False, since } A \subset B \text{ and } x \text{ may not be in } A \end{align}

(a)(ii)

\begin{align} & \text{True, since } B \subset C \end{align}

(b)(i)

$$ A \subset C $$

(b)(ii)

$$ n(A) < n(B) < n(C) $$

(a)

\begin{align} R & = \{ \text{red, orange, yellow, green, blue, indigo, violet} \} \end{align}

(b)

\begin{align} \text{white } \notin R \end{align}

(c)

\begin{align} P \not \subset R \end{align}

(d)

\begin{align} \xi & = \{x : x \text{ is a colour} \} \end{align}

(a)(i)

\begin{align} F & = \{ \text{chicken, fish} \} \end{align}

(a)(ii)

\begin{align} D & = \{ \text{coffee, tea, fruit juice} \} \end{align}

(b)

$$ F \subset E $$

(a)

\begin{align} V \subset Z \text{ and } W \subset Z & \implies Z \text{ contains } 1, 3, 4, 5, 6, 8 \\ \\ X \not \subset Z & \implies Z \text{ do not contain 2 or 4 or 7} \\ \\ Z \not \subset Y & \implies Z = Y \text{ or } n(Z) \ge n(Y) \\ \\ \therefore \text{Minimum value of } n(Z) & = 7 \end{align}

(b)

\begin{align} Z & = \{ 1, 3, 4, 5, 6, 7, 8 \} \phantom{000000} [Z = Y \text{ such that } Z \not \subset Y] \end{align}

(a)

\begin{align} A & = \{ \text{mushrooms, bacon, pineapple} \} \end{align}

(b)

\begin{align} B & = \{ \text{mushrooms, bacon, pineapple, onions, sausage} \} \end{align}

(a)(i)

\begin{align} P & = \{ \text{M1, M2, M3, M4} \} \end{align}

(a)(ii)

\begin{align} Q & = \{ \text{M1, M5} \} \end{align}

(a)(iii)

\begin{align} R & = \{ \text{M5, M6} \} \end{align}

(b)

\begin{align} P & = \{ \text{M1, M2, M3, M4} \} \\ \\ R & = \{ \text{M5, M6} \} \\ \\ \therefore P & \not \subset R \end{align}

(c)

$$ \text{No common elements} $$

\begin{align} A & = \{ 1, 2 \} \\ \\ B & = \{ 1, 2, 3 \} \phantom{0000000} [A \subset B] \\ \\ C & = \{ 1, 2, 4 \} \phantom{0000000} [A \subset C, B \not \subset C, C \not \subset B] \end{align}