(i)

\begin{align*} A & = \{ 2, 3, 5, 7 \} \\ \\ B & = \{3, 5, 7, 9 \} \\ \\ A \cup B & = \{ 2, 3, 5, 7, 9 \} \phantom{000000} [\text{Elements in } A \text{ or } B \text{ or both}] \end{align*}

(ii)

\begin{align*} A & = \{ 2, 3, 5, 7 \} \\ \\ B & = \{3, 5, 7, 9 \} \\ \\ B' & = \{ 2, 4, 6, 8 \} \\ \\ A \cap B' & = \{ 2 \} \end{align*}

(i)

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

(ii)

\begin{align*} A & = \{ \text{a, e} \} \\ \\ A \cup B & = \{ \text{a, c, e, f} \} \phantom{000000} [\text{Alphabets in } A \text{ or } B \text{ or both}] \end{align*}

(iii)

\begin{align*} A & = \{ \text{a, e} \} \\ \\ B' & = \{ \text{b, d, e} \} \\ \\ A \cap B' & = \{ \text{e} \} \end{align*}

(i)

\begin{align*} A & = \{ 2, 3, 5, 7 \} \\ \\ B & = \{ 2, 4, 6, 8, 10 \} \\ \\ C & = \{ 3, 6, 9 \} \\ \\ A \cap B & = \{ 2 \} \end{align*}

(ii)

\begin{align*} B' & = \{ 1, 3, 5, 7, 9 \} \\ \\ A \cup B' & = \{ 1, 2, 3, 5, 7, 9 \} \phantom{000000} [\text{Elements in } A \text{ or } B' \text{ or both}] \end{align*}

(iii)

\begin{align*} C' & = \{ 1, 2, 4, 5, 7, 8, 10 \} \\ \\ A \cap C' & = \{ 2, 5, 7 \} \end{align*}

(i)

\begin{align*} A & = \{ 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32 \} \end{align*}

(ii)

\begin{align*} B & = \{ 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33 \} \end{align*}

(iii)

\begin{align*} C & = \{ 11, 13, 17, 19, 23, 29, 31 \} \end{align*}

(iv)

\begin{align*} A \cap B & = \{ 15, 17, 19, 21, 23, 25, 27, 29, 31 \} \end{align*}

(v)

\begin{align*} B \cap C & = \{ 11, 13, 17, 19, 23, 29, 31 \} \end{align*}

(vi)

\begin{align*} C' & = \{ 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33 \} \\ \\ B \cap C' & = \{ 15, 21, 25, 27, 33 \} \end{align*}

(i)

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

(ii)

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

(i)

\begin{align*} C & = \{ 64, 81, 100 \} \phantom{000000} [\text{Perfect square (i.e. can be square root)}] \\ \\ A \cap C & = \{ 64, 81, 100 \} \end{align*}

(ii)

\begin{align*} B & = \{ \sqrt{64}, \sqrt{81}, \sqrt{100} \} \\ & = \{ 8, 9, 10 \} \end{align*}

(i)

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

(ii)

\begin{align*} A' & = \{ 5, 7, 8, 9, 10, 11 \} \\ \\ B & = \{ 1, 9 \} \phantom{000000} [\sqrt{1} = 1, \sqrt{9} = 9] \\ \\ A' \cap B & = \{ 9 \} \end{align*}

(iii)

\begin{align*} A \cup B & = \{ 1, 2, 3, 4, 6, 9, 12 \} \end{align*}

(a)

(b)(i)

\begin{align*} A \cap B & = \{ 2, 4, 14 \} \end{align*}

(b)(ii)

\begin{align*} A' \cup B & = \{ 1, 2, 3, 4, 5, 7, 9, 11, 13, 14, 15, 17, 19 \} \end{align*}

(b)(iii)

\begin{align*} A' \cap B' & = \{ 3, 5, 9, 11, 13, 15, 17, 19 \} \end{align*}

(i)

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

(ii)

\begin{align*} A & = \{ 2, 3, 5, 7, 11, 13, 17, 19, 23 \} \\ \\ A \cap C & = \{ 2, 3, \} \end{align*}

(iii)

\begin{align*} B & = \{ 3, 6, 9, 12, 15, 18, 21, 24 \} \\ \\ B' & = \{ 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23 \} \\ \\ B' \cap C & = \{ 1, 2, 4, 8 \} \end{align*}

(iv)

\begin{align*} A \cap B' & = \{ 2, 5, 7, 11, 13, 17, 19, 23 \} \end{align*}

(i)

(ii)

Note: $ P \cap Q \ne \emptyset$ means that sets $P$ and $Q$ do intersect!

(i)

(ii)

(a)

$$ A \cap B' $$

(b)

$$ (A \cup B)' $$

(c)

$$ A \cap B' $$

(d)

$$ (A \cap B)' $$

(i) & (iii)

(ii)

\begin{align*} A \subset B \end{align*}

(i)

\begin{align*} 14 - x & > 3 \\ -x & > 3 - 14 \\ -x & > - 11 \\ x & < 11 \\ \\ A & = \{ -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \} \\ \\ 1 - 2x & < 10 \\ -2x & < 10 - 1 \\ -2x & < 9 \\ x & > {9 \over -2} \\ x & > -4.5 \\ \\ B & = \{ -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 \} \\ \\ A \cup B & = \{ -10, -9, ..., 14, 15 \} \\ \\ (A \cup B)' & = \{ \} \end{align*}

(ii)

\begin{align*} A \cap B & = \{ -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \} \end{align*}

(iii)

\begin{align*} B' & = \{ -10, -9, -8, -7, -6, -5 \} \\ \\ A \cap B' & = \{ -10, -9, -8, -7, -6, -5 \} \end{align*}

(a)(i)

$$ \text{No, since rational numbers such as } {3 \over 5} \text{ are not integers.} $$

(a)(ii)

$$ \text{No, since irrational numbers such as } \pi \text{ are real numbers.} $$

(c)

(c)(i)

\begin{align*} A' & = \{ x: x \text{ is an irrational number} \} \\ \\ & \text{Element: } \pi \end{align*}

(c)(ii)

\begin{align*} B' & = \{ x: x \text{ is not an integer} \} \\ \\ A \cap B' & = \{ x: x \text{ is not an integer that can be expressed as a fraction} \} \\ \\ & \text{Element: } {1 \over 3} \end{align*}

(a)

\begin{align*} \{ 2, 4 \} \subset A \phantom{000000} [ \{ 2, 4\} \text{ is a set}] \end{align*}

(b)

\begin{align*} A \cap (B \cup C) & = \{ 6 , 12, 20 \} \\ \\ 9 & \notin A \cap (B \cup C) \end{align*}

(c)

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

(d)

\begin{align*} A \cup B \cup C & = \xi \end{align*}

(a)

\begin{align*} \text{P(Student does not own cat or dog)} & = { 7 \over 7 + 4x + 3x - 6 + x^2 + 3x} \\ {7 \over 40} & = {7 \over x^2 + 10x + 1} \\ \\ 40 & = x^2 + 10x + 1 \\ 0 & = x^2 + 10x + 1 - 40 \\ 0 & = x^2 + 10x - 39 \\ 0 & = (x + 13)(x - 3) \end{align*} \begin{align*} x + 13 & = 0 && \text{ or } & x - 3 & = 0 \\ x & = -13 \text{ (Reject)} &&& x & = 3 \end{align*}

(b)

\begin{align*} n (C \cap D) & = 3x - 6 \\ & = 3(3) - 6 \\ & = 3 \end{align*}

(c)

\begin{align*} \text{No. of students who own a dog only} & = x^2 + 3x \\ & = (3)^2 + 3(3) \\ & = 18 \end{align*}