(a)

\begin{align} A \cup B & = \{ 1, 3, 6, 7, 8, 9, 12 \} \end{align}

(b)

\begin{align} A \cup C & = \{ 3, 4, 6, 8, 9, 12 \} \end{align}

(c)

\begin{align} B \cup C & = \{ 1, 3, 4, 6, 7, 8, 9 \} \end{align}

(a)

\begin{align} D \cap E & = \{ \text{b, d} \} \end{align}

(b)

\begin{align} D \cap F & = \{ \text{} \} \end{align}

(c)

\begin{align} E \cap F & = \{ \text{g} \} \end{align}

(a)

\begin{align} X \cap Y & = \{ \text{line} \} \end{align}

(b)

\begin{align} X \cap Z & = \{ \text{line, ray} \} \end{align}

(c)

\begin{align} Y \cap Z & = \{ \text{line} \} \end{align}

(a)

\begin{align} P \cup Q & = \{ \text{hop, jog, run, walk} \} \end{align}

(b)

\begin{align} P \cup R & = \{ \text{hop, jog, run, walk, skip} \} \end{align}

(c)

\begin{align} Q \cup R & = \{ \text{jog, walk, hop, run, skip} \} \end{align}

(a)

\begin{align} 2x & < 9 \\ x & < {9 \over 2} \\ x & < 4.5 \\ \\ P & = \{ 1, 2, 3, 4 \} \\ \\ Q & = \{ -3, -2, -1, 0, 1, 2 \} \\ \\ P \cup Q & = \{ -3, -2, -1, 0, 1, 2, 3, 4 \} \end{align}

(b)

\begin{align} P \cap Q & = \{ 1, 2 \} \end{align}

(a)

\begin{align} X' & = \{ \text{joy, love, trust} \} \end{align}

(b)

\begin{align} X \cup X' & = \{ \text{care, joy, love, peace, trust} \} \end{align}

(c)

\begin{align} X \cap X' & = \{ \text{} \} \end{align}

(d)

\begin{align} X \cup \xi & = \xi = \{ \text{care, joy, love, peace, trust} \} \end{align}

(e)

\begin{align} X \cap \xi & = X = \{ \text{care, peace} \} \end{align}

(a)

\begin{align} -9 & < 2x < 6 \\ -4.5 & < x < 3 \\ \\ \xi & = \{ -4, -3, -2, -1, 0, 1, 2 \} \\ \\ \\ x^2 - 4 & = 0 \\ x^2 & = 4 \\ x & = \pm \sqrt{4} \\ x & = \pm 2 \\ \\ A & = \{ -2, 2 \} \\ \\ \\ 1 - 3x & \le 7 \\ -3x & \le 7 - 1 \\ -3x & \le 6 \\ x & \ge {6 \over -3} \\ x & \ge -2 \\ \\ B & = \{ -2, -1, 0, 1, 2 \} \\ \\ \\ \therefore A & \subset B \end{align}

(b)(i)

\begin{align} A \cup B & = B = \{ -2, -1, 0, 1, 2 \} \end{align}

(b)(ii)

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

(a)

(b)(i)

\begin{align} A \cup B & = \{ \text{r, f, d, m, s} \} \end{align}

(b)(ii)

\begin{align} (A \cup B)' & = \{ \text{l, t} \} \end{align}

(b)(iii)

\begin{align} A \cap B & = \{ \text{d, m} \} \end{align}

(b)(iv)

\begin{align} (A \cap B)' & = \{ \text{r, f, s, l, t} \} \end{align}

(b)(v)

\begin{align} A' \cup B' & = \{ \text{r, f, s, l, t} \} \end{align}

(b)(vi)

\begin{align} A' \cap B' & = \{ \text{l, t} \} \end{align}

(c)

\begin{align} (A \cup B)' = A' \cap B' \text{ and } (A \cap B)' = A' \cup B' \end{align}

(a)

\begin{align} Q & = \{ 8^2 , 9^2 , 10^2 \} \\ & = \{ 64, 81, 100 \} \\ \\ \text{Statement is false} & \text{ since the element 81 is not a multiple of 4} \end{align}

(b)

\begin{align} Q & = \{ 64, 81, 100 \} \\ \\ R & = \{ 4^3 \} = \{ 64 \} \\ \\ Q \cap R & = \{ 64 \} \text{ and statement is false} \end{align}

(c)

\begin{align} R & = \{ 64 \} \\ \\ \text{Since 64 is a } & \text{multiple of 4, } R \subset P \\ \\ \therefore P \cup R & = P \text{ is true} \end{align}

(d)

\begin{align} \text{False since } 60 \notin \xi \end{align}

(a)

(b)(i)

\begin{align} X \cup Y & = \{ \text{n, t, m, p, r, u} \} \end{align}

(b)(ii)

\begin{align} X' \cup Y' & = \{ \text{n, t, u, q, s} \} \end{align}

(b)(iii)

\begin{align} X' \cap Y' & = \{ \text{q, s} \} \end{align}

(b)(iv)

\begin{align} X' \cap Y & = \{ \text{u} \} \end{align}

(a)

(b)(i)

\begin{align} A \cup B' & = \{ 2, 3, 5, 7, 11, 13, 8, 9, 10 \} \end{align}

(b)(ii)

\begin{align} A' \cap B & = \{ 1, 4, 6, 12 \} \end{align}

(a)(i)

(a)(ii)

(a)(iii)

(b)

\begin{align} A \cap B = \emptyset \text{ and } B \subset A' \text{ are equivalent} \end{align}

(a)

(b)

(c)

(d)

(a)(i)

(a)(ii)

(b)

\begin{align} \text{For (a)(i), } A \cap B & = A \\ \\ \text{For (a)(ii), } A \cup B & = B \end{align}

(a)(i)

(a)(ii)

(a)(iii)

(a)(iv)

(b)

(a)

\begin{align} \xi & = \{ 1, 2, 3, 4, 5, 6, 7, 8, ... \} \\ \\ A & = \{ 4, 8, 12, 16, 20, ... \} \\ \\ B & = \{ 6, 12, 18, 24, 30, ... \} \end{align}

(b)(i)

\begin{align} A \cup B & = \{ 4, 6, 8, 12, 16, 18, ... \} \end{align}

(b)(ii)

\begin{align} A \cap B & = \{ 12, 24, 36, 48, 60, ... \} \phantom{000000} [\text{Lowest common multiple of 4 and 6 is 12}] \end{align}

(c)

\begin{align} A \cap B & = \{ x : x \text{ is a multiple of 12} \} \end{align}

(a)

$$ X' $$

(b)

\begin{align} \underbrace{ (P \cup Q)' }_\text{'Outside part'} \cup (P \cap Q) \end{align}

(a)

\begin{align} \xi & = \{ \text{Days in a week} \} \end{align}

(b)

\begin{align} A \cup B & = \{ \text{Monday, Tuesday, Wednesday, Friday, Saturday, Sunday} \} \\ \\ \text{It re} & \text{presents the days of the week that have at least one helper} \end{align}

(c)

\begin{align} A \cap B & = \{ \text{Saturday, Sunday} \} \\ \\ \text{It re} & \text{presents the days of the week that have two helpers} \end{align}

(d)

\begin{align} (A \cup B)' & = \{ \text{Thursday} \} \\ \\ \text{Assume the stall needs at least} & \text{ one helper to operate, } \text{stall does not operate on Thursday} \end{align}

(a)

\begin{align} X \cap Y & = \{ \text{Ben} \} \\ \\ \text{Ben is the only brand } & \text{preferred by both Mr and Mrs Singh} \end{align}

(b)

\begin{align} X' \cap Y' & = \{ \text{Mit, Nis} \} \\ \\ \text{The two brands that are not } & \text{preferred by both Mr and Mrs Singh} \end{align}

(c)

\begin{align} \text{Assuming other family member(s) have no say, the couple will buy Ben} \end{align}

(a)

\begin{align} G \cap S' \text{ represents the students in uniformed group but not in sports team} \end{align}

(b)(i)

\begin{align} \text{No. of students} & = 7 \end{align}

(b)(ii)

\begin{align} \text{No. of students} & = 38 - 9 - 7 - 5 \\ & = 17 \end{align}

(a)

\begin{align} \text{Let } & y = 0, \\ 21 - x + x + 19 - x & = 36 \\ -x + x - x & = 36 - 21 - 19 \\ -x & = -4 \\ x & = 4 \end{align}

\begin{align} n(R \cap S) & = n(S) = 19 \\ \\ \\ \therefore \text{Max.} & = 19, \text{Min.} = 4 \end{align}

(b)

\begin{align} n ( R \cap S) & = 10 \end{align}

\begin{align} A & = \{ 3, 6, 1, 2 \} \\ \\ B & = \{ 3, 6, 4, 5 \} \end{align}