(i)

\begin{align} \xi & = \{ \text{cat, dog, mouse, lion, tiger} \} \\ \\ A & = \{ \text{cat, dog, mouse} \} \end{align}

(ii)

\begin{align} A' & = \{ \text{lion, tiger} \} \end{align}

(i)

\begin{align} \xi & = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \} \\ \\ B & = \{ 2, 4, 6, 8, 10 \} \end{align}

(ii)

(iii)

\begin{align} B' & = \{ 1, 3, 5, 7, 9 \} \end{align}

(iv)

\begin{align} B' \text{ is the set of positive odd integers between 1 and 10 inclusive} \end{align}

(i)

\begin{align} C & = \{ \text{s, t, u} \} \\ \\ D & = \{ \text{s, t, u, v, w, x, y, z} \} \end{align}

(ii)

\begin{align} \text{Yes, since all elements in } C \text{ are in set } D \text{ and there are more elements in } D \text{ than in } C \end{align}

(a)

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

(b)

\begin{align} \text{True} \phantom{000000} [\text{The superset must have more elements than the subset}] \end{align}

(c)

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

(d)

\begin{align} \text{False} \end{align}

(a)

\begin{align} \{ \}, \{ \text{a} \}, \{ \text{b} \}, \{ \text{a, b} \} \end{align}

(b)

\begin{align} \{ \}, \{ \text{Singapore} \}, \{ \text{Malaysia} \}, \{ \text{Singapore, Malaysia} \} \end{align}

(c)

\begin{align} \{ \}, \{ 14 \}, \{ 16 \}, \{ 14,16 \} \end{align}

(d)

\begin{align} \{ \}, \{ 7 \} \end{align}

(i)

\begin{align} \xi & = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \} \\ \\ J' & = \{ 2, 3, 5, 7 \} \\ \\ J & = \{ 1, 4, 6, 8, 9 \} \end{align}

(ii)

\begin{align} J' \text{ is the set of the prime numbers between 0 and 10 } \end{align}

(iii)

\begin{align} n(\xi) & = 9 \\ \\ n(J) & = 5 \\ \\ n(J') & = 4 \end{align}

(iv)

\begin{align} \text{Yes, since elements in } J \text{ and } J' \text{ are in the universal set } \xi \end{align}

(i)

\begin{align} \xi & = \{ \text{a, b, c, d, e, f, g, h, i, j} \} \\ \\ K & = \{ \text{b, c, d, f, g, h, j} \} \phantom{000000} [\text{Consonants: Non-vowels (letter except A E I O U)}] \\ \\ K' & = \{ \text{a, e, i} \} \end{align}

(ii)

\begin{align} K' \text{ is the set of vowels from the first 10 letters of the English alphabet} \end{align}

(iii)

\begin{align} n(\xi) & = 10 \\ \\ n(K) & = 7 \\ \\ n(K') & = 3 \end{align}

(iv)

\begin{align} \text{Yes, since elements in } K \text{ and } K' \text{ are in the universal set } \xi \end{align}

(i)

(ii)

\begin{align} \text{Yes, since all elements in } M \text{ are in } L \text{ and there are more elements in } L \text{ than in } M \end{align}

(iii)

\begin{align} \text{Yes, since not all elements in } L \text{ are in } M \end{align}

(i)

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

(ii)

\begin{align} P & \subset N \\ \\ \text{Every element in } P \text{ is in } N & \text{ and there are elements in } N \text{ that are not in } P \end{align}

(iii)

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

(iv)

\begin{align} N \subseteq Q & \text{ and } Q \subseteq N \\ \\ \text{Every element in } N & \text{ is in } Q \text{ (and vice versa)} \end{align}

(a)

\begin{align} \{ \}, \{ 7 \}, \{ 8 \}, \{ 9 \}, \{ 7, 8 \}, \{ 8, 9 \} , \{ 7, 9 \}, \{ 7, 8, 9 \} \end{align}

(b)

\begin{align} S & = \{ 2, 3, 5 \} \\ \\ \text{Possible subsets: } & \{ \}, \{ 2 \}, \{ 3 \}, \{ 5 \}, \{ 2, 3 \}, \{ 3, 5 \}, \{ 2, 5 \}, \{ 2, 3, 5 \} \end{align}

(c)

\begin{align} \{ \}, \{ \text{a} \}, \{ \text{b} \}, \{ \text{c} \}, \{ \text{d} \}, \{ \text{a, b} \}, \{ \text{b, c} \}, \{ \text{c, d} \} , \{ \text{a, c} \}, \{ \text{a, d} \}, \{ \text{b, d} \}, \{ \text{a, b, c} \}, \{ \text{a, b, d} \}, \{ \text{a, c, d} \}, \{ \text{b, c, d} \}, \{ \text{a, b, c, d} \} \end{align}

(d)

\begin{align} U & = \{ \text{U, N, I, O} \} \\ \\ \text{Possible subsets: } & \{ \}, \{ \text{U} \}, \{ \text{N} \}, \{ \text{I} \}, \{ \text{O} \}, \{ \text{U, N} \}, \{ \text{U, I} \}, \{ \text{U, O} \}, \{ \text{N, I} \}, \{ \text{N, O} \}, \{ \text{I, O} \}, \\ & \{ \text{U, N, I} \}, \{ \text{U, N, O} \}, \{ \text{U, I, O} \}, \{ \text{N, I, O} \}, \{ \text{U, N, I, O} \} \end{align}

(i)

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

(ii)

\begin{align} V' \text{ is the set of positive integers less than 21 that are not multiples of 3} \end{align}

(i)

\begin{align} & [\text{Rational numbers are numbers that can be expressed as a fraction}] \\ \\ & \text{Yes, since all integers can be expressed as fractions and there are some rational numbers that are not integers (i.e. } 1{1 \over 2}) \end{align}

(ii)

\begin{align} & \text{Yes, there are rational numbers that are not integers, i.e. } 1{1 \over 2}, 10.5 \end{align}

\begin{align} \text{Consider } A & = \{ 1 \} \\ \\ n(A) & = 1 \\ \\ \text{Possible subsets: } & \{ \}, \{ 1 \} \phantom{000000} [\text{Total 2}] \\ \\ \\ \text{Consider } B & = \{ 1, 2 \} \\ \\ n(B) & = 2 \\ \\ \text{Possible subsets: } & \{ \}, \{ 1 \}, \{ 2 \}, \{ 1, 2 \} \phantom{000000} [\text{Total } 4 = 2^2] \\ \\ \\ \text{Consider } C & = \{ 1, 2, 3 \} \\ \\ n(C) & = 3 \\ \\ \text{Possible subsets: } & \{ \}, \{ 1 \}, \{ 2 \}, \{ 3 \}, \{1, 2\}, \{1, 3\}, \{ 2, 3 \}, \{ 1, 2, 3 \} \phantom{000000} [\text{Total } 8 = 2^3 ] \\ \\ \\ \\ \therefore \text{Number of subsets of } Y & = 2^a \end{align}