(a)

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

(b)

\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)

$$ B' \text{ is the set of positive odd integers that are smaller than the integer } 10 $$

(i)

\begin{align*} A & = \{ \text{s, t, u} \} \\ \\ B & = \{ \text{s, t, u, v, w, x, y, z} \} \end{align*}

(ii)

$$ A \text{ is a proper subset of } B, \text{ since every element of } A \text{ is an element of } B \text{ and } B \text{ contains elements that are not in } A $$

(i)

(ii)

$$ B \text{ is a proper subset of } A, \text{ since every element of } B \text{ is an element of } A \text{ and } A \text{ contains elements that are not in } B $$

(i)

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

(ii)

$$ C' \text{ is the set of prime numbers lesser than } 10 $$

(a) Consonants are all the letters excluding vowels (a, e, i, o, u)

\begin{align*} \xi & = \{ \text{a, b, c, d, e, f, g, h, i, j} \} \\ \\ D & = \{ \text{b, c, d, f, g, h, j} \} \\ \\ D' & = \{ \text{a, e, i} \} \end{align*}

(b)

$$ D' \text{ is the set of vowels limited to the first 10 letters of the English alphabet} $$

(i)

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

(ii)

$$ F \subset E $$ $$ \text{Every element of } F \text{ is an element of } E \text{ and } E \text{ contains elements that are not in } F $$

(iii)

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

(iv)

$$ E = G $$

$$ \text{Yes, } I \text{ is a proper subset of } H $$ $$ \text{Every integer is a rational number, i.e. } 5 = {5 \over 1}$$ $$ \text{At least one rational number is not an integer, i.e. } 1{1 \over 10} $$

(a)

$$ \{ \}, \{ x \}, \{ y \} $$

(b)

$$ \{ \}, \{ \text{Singapore} \}, \{ \text{Malaysia} \} $$

(c)

$$ \{ \}, \{ 3 \}, \{ 4 \}, \{ 5 \}, \{ 3, 4 \}, \{4, 5 \}, \{3, 5\} $$

(d)

$$ \{ \}, \{ \text{a} \}, \{ \text{b} \}, \{ \text{c} \}, \{ \text{d} \}, \{ \text{a, b} \}, \{ \text{a, c} \}, \{ \text{a, d} \}, \{ \text{b, c} \}, \{ \text{b, d} \}, \{ \text{c, d} \}, \{ \text{a, b, c} \}, \{ \text{a, c, d} \}, \{ \text{a, b, d} \}, \{ \text{b, c, d} \} $$

(i)

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

(ii)

\begin{align*} O' & = \{ x: x \text{ is an integer not divisible by 3 and } 1 \le x < 21 \} \end{align*}