(i)

\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}

(ii)

(iii)(a)

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

(iii)(b)

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

(i)

\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}

(ii)

(iii)(a)

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

(iii)(b)

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

(i)

\begin{align} \xi & = \{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 \} \\ \\ P & = \{ 2, 3, 5, 7, 11 \} \\ \\ Q & = \{ 4, 6, 8, 9, 10 \} \phantom{000000} [\text{Composite number has more than 2 factors}] \end{align}

(ii)

(iii)(a)

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

(iii)(b)

\begin{align} (P \cup Q)' & = \{ 0, 1 \} \end{align}

(iii)(c)

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

(i)

(ii)

(i)

(a)

$$ A $$

(b)

$$ B $$

(c)

$$ A \cap B $$

(d)

$$ A \cup B $$

(e)

$$ A' $$

(f)

$$ B' $$

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(Yeah it's blank!)

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(a)

$$ X \cap Y' $$

(b)

$$ X' \cap Y $$

(c)

$$ (X \cup Y)' $$

(d)

$$ (X \cap Y)' $$

(e)

$$ X \cup Y' $$

(f)

$$ X' \cup Y $$

(g)

$$ X \cap Y' $$

(h)

$$ X' \cup Y $$

(i)

\begin{align} n(A \cup B)' & = 17 - 4 - 8 \\ & = 5 \end{align}

(ii)

\begin{align} n(A \cup B)' & = 0 \end{align}

(i)

\begin{align} n(X \cap Y)' & = 8 + 8 \\ & = 16 \end{align}

(ii)

\begin{align} n(X \cap Y)' & = 25 - 1 \\ & = 24 \end{align}

(i)

\begin{align} n(A \cup B)' & = 10 \end{align}

(ii)

\begin{align} n(A \cup B)' & = 1 \end{align}

(i)

\begin{align} n (X \cap Y)' & = 3 + 4 \\ & = 7 \end{align}

(ii)

\begin{align} n (X \cap Y)' & = 4 + 7 \\ & = 11 \end{align}

(i)

\begin{align} A \cap \xi & = A \end{align}

(ii)

\begin{align} A \cup \xi & = \xi \end{align}

(iii)

\begin{align} A \cap \emptyset & = \emptyset \end{align}

(iv)

\begin{align} A \cup \emptyset & = A \end{align}

(i)

\begin{align} A \cap B & = A \phantom{000000} [\text{Since } A \text{ is a proper subset of } B] \end{align}

(ii)

\begin{align} A \cup B & = B \phantom{000000} [\text{Since } A \text{ is a proper subset of } B] \end{align}

(iii)

\begin{align} \text{Not possible to simplify} \end{align}

(iv)

\begin{align} \text{Not possible to simplify} \end{align}

(v)

\begin{align} (B \cup C) \cap A & = A \end{align}

(vi)

\begin{align} (B \cap C) \cap A & = \emptyset \end{align}

(vii)

\begin{align} \text{Not possible to simplify} \end{align}

(viii)

\begin{align} (A \cap C) \cup B & = B \end{align}

(i)

\begin{align} (P \cap Q') \cup (P' \cap Q) \end{align}

(ii)

\begin{align} [(P \cap Q') \cup (P' \cap Q)] \cup (P' \cap Q') \end{align}

(i)

\begin{align} \text{No. of students} & = 15 + 9 \\ & = 24 \end{align}

(ii)

\begin{align} \text{No. of students} & = 38 - 15 - 9 \phantom{000000} [\text{Refer to Venn diagram in (i)}] \\ & = 14 \end{align}

(iii)

\begin{align} \text{No. of students} & = 9 \end{align}

(iv)

\begin{align} \text{No. of students} & = 0 \phantom{000000} [\text{Refer to Venn diagram in iii - all students} \\ & \phantom{0000000000} \text{who enjoy swimming enjoy cycling as well!}] \end{align}

(i)

\begin{align} \text{No. of children} & = 6 \end{align}

(ii)

\begin{align} \text{No. of children} & = 35 - 7 - 6 - 22 \phantom{000000} [\text{Refer to Venn diagram in i}] \\ & = 0 \end{align}

(ii)

\begin{align} \text{No. of children} & = 15 + 13 \\ & = 28 \end{align}

(iv)

\begin{align} \text{No. of children} & = 15 \phantom{000000} [\text{Refer to Venn diagram in iii}] \end{align}