(a)

\begin{align} \text{pen} \notin A \end{align}

(b)

\begin{align} \text{Yes, } \{ \text{marker} \} \text{ is a proper subset of } A \end{align}

(c)

\begin{align} \xi & = \{ \text{stationaries} \} \end{align}

(d)

\begin{align} B & = \{ \text{ruler, pencil, marker} \} \end{align}

(a)(i)

\begin{align} A & = \{ \text{l, a, t, e, r} \} \end{align}

(a)(ii)

\begin{align} B & = \{ \text{l, a, t, e, r} \} \end{align}

(a)(iii)

\begin{align} C & = \{ \text{l, t, e, r} \} \end{align}

(b)(i)

\begin{align} n(A) & = 5 \end{align}

(b)(ii)

\begin{align} n(B) & = 5 \end{align}

(b)(iii)

\begin{align} n(C) & = 4 \end{align}

(c)(i)

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

(c)(ii)

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

(a)(i)

\begin{align} \xi & = \{ \text{m, a, e, t, k, p, g, d, n, y} \} \end{align}

(a)(ii)

\begin{align} A & = \{ \text{m, a, e, t} \} \end{align}

(a)(iii)

\begin{align} B' & = \{ \text{m, a, d, n, y} \} \end{align}

(a)(iv)

\begin{align} A \cap B & = \{ \text{e, t} \} \end{align}

(b)

\begin{align} \{ \text{m, a} \} \notin A \phantom{00000} [ \{ \text{m, a} \} \text{ is a set} ] \end{align}

(c)

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

(d)

\begin{align} D & = \{ \text{k, p, g} \} \end{align}

(a)(i)

\begin{align} P & = \{ 1, 2, 3, 4, 6, 12 \} \end{align}

(a)(ii)

\begin{align} P \cup Q & = \{ 1, 2, 3, 4, 5, 6, 12 \} \end{align}

(b) Note: My answer is different from the answer key - Since universal set is not defined, I'm assuming that the elements in set R includes zero and negative integers.

\begin{align} 2x & < 12 \\ x & < {12 \over 2} \\ x & < 6 \\ \\ R & = \{ 5, 4, 3, 2, 1, 0, -1, ... \} \\ \\ \therefore R \not \subset Q & \text{ since some elements in } R \text{ are not in } Q \end{align}

(a)

\begin{align} \xi & = \{ \text{quadrilaterals} \} \end{align}

(b)

\begin{align} A \cap B & = \{ \text{square} \} \end{align}

(c)

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

(d)

\begin{align} \text{trapezium} & \notin C \text{ (since trapezium has only 1 pair of parallel sides)} \end{align}

(a)

\begin{align} \text{No right-angled triangle has 3 equal sides} \end{align}

(b) Note: A scalene triangle has all sides of different lengths

\begin{align} & \text{Since } P \in \xi, \text{ triangle } P \text{ has a right-angle} \\ \\ & \text{Since } P \in X' \text{ and } Y = \emptyset, \text{ triangle } P \text{ is isosceles} \\ \\ & \text{Acute angle in } P = {180^\circ - 90^\circ \over 2} = 45^\circ \end{align}

(a)(i)

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

(a)(ii)

\begin{align} B & = \{ 1^2, 2^2, 3^2, 4^2 \} \\ B & = \{ 1, 4, 9, 16 \} \\ \\ A \cap B' & = \{ 3, 5, 7, 11, 13, 15, 17, 19 \} \end{align}

(a)(iii)

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

(b)

\begin{align} A \cap B \text{ is the set of odd integers that are perfect squares} \end{align}

(a)

\begin{align} A & = \{ 2, 3, 5, 7, 11, 13 \} \end{align}

(b)

\begin{align} B & = \{ 3, 6, 9, 12 \} \\ \\ B' & = \{ 2, 4, 5, 7, 8, 10, 11, 13, 14 \} \end{align}

(c)

\begin{align} A \cap B' & = \{ 2, 5, 7, 11, 13 \} \end{align}

(d)

\begin{align} A \cup B' & = \{ 2, 3, 4, 5, 7, 8, 10, 11, 13, 14 \} \\ \\ (A \cup B')' & = \{ 6, 9, 12 \} \end{align}

(a)

\begin{align} 3x - 1 & < 2 \\ 3x & < 2 + 1 \\ 3x & < 3 \\ x & < 1 \\ \\ P & = \{ -4, -3, -2, -1, 0 \} \\ \\ P' & = \{ 1, 2 \} \end{align}

(b)

\begin{align} 2x^2 + 7x + 3 & = 0 \\ (2x + 1)(x + 3) & = 0 \end{align} \begin{align} 2x + 1 & = 0 && \text{ or } & x + 3 & = 0 \\ 2x & = -1 &&& x & = -3 \\ x & = -{1 \over 2} \end{align}
\begin{align} Q & = \{ -3 \} \\ \\ P \cap Q & = \{ -3 \} \end{align}

(c)

\begin{align} Q' & = \{ -4, -2, -1, 0, 1, 2 \} \\ \\ P \cup Q' & = \{ -4, -3, -2, -1, 0, 1, 2 \} \end{align}

(a)(i)

(a)(ii)

(b)(i)

(b)(ii)

(a)

\begin{align} ( A \cup B)' \end{align}

(b)

\begin{align} A' \cap B \end{align}

(a)

(b)(i)

(b)(ii)

(b)(iii)

(b)(iv)

(a)(i)

\begin{align} M' & = \{ \text{b, v, x, y, z} \} \end{align}

(a)(ii)

\begin{align} N' & = \{ \text{a, b, u, v} \} \end{align}

(a)(iii)

\begin{align} M \cup N & = \{ \text{a, c, t, u, x, y, z} \} \end{align}

(a)(iv)

\begin{align} M \cap N & = \{ \text{c, t} \} \end{align}

(a)(v)

\begin{align} M \cup N' & = \{ \text{a, b, c, t, u, v} \} \end{align}

(a)(vi)

\begin{align} M' \cap N & = \{ \text{x, y, z} \} \end{align}

(b)

(a)

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

(b)

\begin{align} E \cap N & = \{ \text{City Hall, Raffles Place, Jurong East} \} \end{align}

(c)

\begin{align} & \{ \}, \{ \text{City Hall} \}, \{ \text{Raffles Place} \}, \{ \text{Jurong East} \}, \{ \text{City Hall, Raffles Place} \}, \\ & \{ \text{City Hall, Jurong East} \}, \{ \text{Raffles Place, Jurong East} \} \end{align}