(i)

\begin{align*} A \cap B & = \{ 2, 4 \} \end{align*}

(ii)

(i)

\begin{align*} C \cap D & = \{ \text{blue, yellow, pink} \} \end{align*}

(ii)

(i)

(ii)

\begin{align*} E \cup F & = \{ 11, 12, 13, 14, 15, 16, 17, 18, 20 \} \end{align*}

(i)

(ii)

\begin{align*} G \cup H & = \{ \text{orange, durian, pear, apple, banana, grape, strawberry} \} \end{align*}

(a)

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

(b)

(c)(i)

\begin{align*} (I \cup J)' & = \{ 3, 5, 6, 7, 9, 10, 11, 13, 14, 15 \} \end{align*}

(c)(ii)

\begin{align*} I \cap J' & = \{ 12 \} \end{align*}

(i)

(ii)

(iii)

(i)

\begin{align*} M & = \{ 1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2, 8^2 \} \\ & = \{ 1, 4, 9, 16, 25, 36, 49, 64 \} \\ \\ P & = \{ 1^3, 2^3, 3^3, 4^3 \} \\ & = \{ 1, 8, 27, 64 \} \end{align*}

(ii)

(iii)

\begin{align*} M \cap P & = \{ 1, 64 \} \end{align*}

(i)

\begin{align*} N & = \{ 8, 16, 24, 32 \} \\ \\ Q & = \{ 4, 8, 12, 16, 20, 24, 28, 32 \} \end{align*}

(ii)

\begin{align*} N \cap Q & = \{ 8, 16, 24, 32 \} \end{align*}

(iii)

(iv)

$$ \text{Yes, since } N \text{ is a proper subset of } Q, \text{ every element of } N \text{ is in } Q $$

(i) Composite numbers are numbers with more than 2 factors

\begin{align*} R & = \{ 1, 2, 3, 6, 9, 18 \} \\ \\ S & = \{ 10, 12, 14, 15, 16 \} \end{align*}

(ii)

\begin{align*} R \cap S & = \{ \} \\ \\ \text{Factors } & \text{of 18 are less than or equals to 9 (except 18)} \end{align*}

(iii)

(i)

\begin{align*} T & = \{ 4, 8, 12 \} \\ \\ U & = \{ 1, 2, 3, 4, 6, 8, 12, 24 \} \end{align*}

(ii)

(iii)

\begin{align*} T \cup U & = \{ 1, 2, 3, 4, 6, 8, 12, 24 \} \end{align*}

(iv)

$$ \text{Yes, } T \text{ is a proper subset of } U $$

(i)

\begin{align*} V & = \{ 1, 5, 25 \} \\ \\ W & = \{ 6, 12, 18, 24 \} \end{align*}

(ii)

(iii)

\begin{align*} V \cup W & = \{ 1, 5, 6, 12, 18, 24, 25 \} \end{align*}

(a)

\begin{align*} \xi & = \{4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 \} \\ \\ Y & = \{ 6, 9, 12, 15, 18 \} \\ \\ Z & = \{ 9, 18 \} \end{align*}

(b)

(c)(i)

\begin{align*} (Y \cup Z)' & = \{ 4, 5, 7, 8, 10, 11, 13, 14, 16, 17 \} \end{align*}

(c)(ii)

\begin{align*} Y \cap Z' & = \{ 6, 12, 15 \} \end{align*}

(a)

\begin{align*} \xi & = \{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 \} \\ \\ P & = \{ 2, 3, 5, 7, 11 \} \\ \\ Q & = \{ 0, 1, 4, 6, 8, 9, 10 \} \end{align*}

(b)

(c)(i)

\begin{align*} P \cup Q & = \{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 \} \end{align*}

(c)(ii)

\begin{align*} (P \cup Q)' & = \{ \} \text{ or } (P \cup Q)' = \emptyset \end{align*}

(c)(iii)

\begin{align*} P' \cap Q & = \{ 0, 1, 4, 6, 8, 9, 10 \} \end{align*}

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(i)

$$ A \cap \xi = A $$

(ii)

$$ A \cup \xi = \xi $$

(iii)

$$ A \cap \emptyset = \emptyset $$

(iv)

$$ A \cup \emptyset = A $$

Missing information from the question: Set C is a proper subset of set B

(i)

$$ A \cap B = A $$

(ii)

$$ A \cup B = B $$

(iii)

$$ B \cap C = C $$

(iv)

$$ \text{Not possible to simplify} $$

(v)

$$ (B \cup C) \cap A = B \cap A = A $$

(vi)

$$ (B \cap C) \cap A = C \cap A = \emptyset $$

(vii)

$$ (A \cup C) \cap B = A \cup C $$

(viii)

$$ (A \cap C) \cup B = \emptyset \cup B = B $$