Solutions
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Try it yourself 1
(a)
C={s, t, u, d, e, n}
(b)
n(C)=6
(c)(i)
Yes
(c)(ii)
No
Try it yourself 2
(a)
B={Jan, Feb, Mar, Apr, May, Jun}
(b)
n(B)=6
(c)
June∈B
Try it yourself 3
(a)
R={3,5,7,11,13,17,19}
(b)
S={5,10,15,20}n(S)=4
(c)
Yes, since 5 is a prime number and a multiple of 5
Try it yourself 4
C={s, t, r, e, d }D={d, e, s, r, t }∴
Try it yourself 5
(a)
\begin{align}
x^2 - 2x - 3 & = 0 \\
(x + 1)(x - 3) & = 0
\end{align}
\begin{align}
x + 1 & = 0 && \text{ or } & x - 3 & =0 \\
x & = -1 &&& x & = 3
\end{align}
\begin{align}
P & = \{ -1, 3 \}
\end{align}
(b)(i)
\begin{align}
Q & \not \subset P
\end{align}
(b)(ii)
\begin{align}
P & \subset R
\end{align}
(c)(i)
\begin{align}
S & = \{ -1, 0, 1, 2, 3, 4 \}
\end{align}
(c)(ii)
\begin{align}
n(S) & = 6
\end{align}
Try it yourself 6
\begin{align}
M & = \{ 4, 8, 12, 16, 20, 24, ... \} \\
\\
N & = \{ 12, 24, ... \} \\
\\
\text{Every multiple of 12 is} & \text{ a multiple of 4 (but not vice versa)} \\
\\
\text{Since every element in} & \text{ } N \text{ is in } M \text{ and there are more elements in } M
\text{ than in } N, N \subset M
\end{align}
Try it yourself 7
(a)
\begin{align}
\xi & = \{ x: x \text{ is a tool} \}
\end{align}
(b)
\begin{align}
\xi & = \{ x: x \text{ is an odd number} \}
\end{align}
Try it yourself 8
(a)
\begin{align}
& \text{Since } \emptyset \text{ has no elements, it is a proper subset of } B \\
\\
& \text{Since every element in } B \text{ is in } \xi \text{ and } B \ne \xi, B \subset \xi \\
\\
& \phantom{000000} \therefore \emptyset \subset B \subset \xi
\end{align}
(b)
\begin{align}
\{ \}, \{ 1 \}, \{ 2 \}, \{ 4 \}, \{ 1, 2 \}, \{ 1, 4 \}, \{ 2, 4 \}
\end{align}
Try it yourself 9
(a)(i)
\begin{align}
P & = \{ 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 \}
\end{align}
(a)(ii)
\begin{align}
Q & = \{ 6, 12, 18 \}
\end{align}
(b)(i)
\begin{align}
\text{True, since there are no elements in } \emptyset
\end{align}
(b)(ii)
\begin{align}
\text{False, since every element in } Q \text{ is in } P \text{ and } Q \ne P
\end{align}
(b)(iii)
\begin{align}
\text{True, since } P & = \{ 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 \}
\end{align}
(b)(iv)
\begin{align}
\text{True, since every element in } \{ 12, 18 \} \text{ is in } Q
\text{ and } \{ 12, 18 \} \ne Q
\end{align}
Try it yourself 10
Try it yourself 11
Try it yourself 12
(a)
\begin{align}
B & = \{ 6, 9, 12 \} \\
\\
B' & = \{ 4, 5, 7, 8, 10, 11, 13, 14 \}
\end{align}
(b)
(c)
\begin{align}
B' \text{ is the set of integers between 4 and 14 inclusive and are not multiples of 3}
\end{align}
Try it yourself 13
(a)
\begin{align}
R & = \{ 25, 36 \}
\phantom{000000} [5^2 , 6^2] \\
\\
S & = \{ 26, 28, 30, 32, 34, 36 \}
\end{align}
(b)
\begin{align}
\text{No, since } 25 \notin S
\end{align}
(c)(i)
\begin{align}
R' & = \{ 26, 27, 28, 29, 30, 31, 32, 33, 34, 35 \}
\end{align}
(c)(ii)
\begin{align}
S' & = \{ 25, 27, 29, 31, 33, 35 \}
\end{align}
Try it yourself 14
(a)
\begin{align}
P \cap Q & = \{ \}
\end{align}
(b)
\begin{align}
P \cap R & = \{ 6, 8 \}
\end{align}
(c)
\begin{align}
Q \cap R & = \{ 4 \}
\end{align}
Try it yourself 15
(a)
(b)(i)
(b)(ii)
\begin{align}
P' \cap Q & = \{ e \}
\end{align}
(c)
\therefore P' \cap Q \ne P \cap Q'
Try it yourself 16
(a)
\begin{align}
C \cup D & = \{ \text{b, r, e, a, t, h, d, i, w} \}
\end{align}
(b)
\begin{align}
\text{Yes, as all elements in } D \text{ is in } C \cup D \text{ and } D \ne (C \cup D)
\end{align}
Try it yourself 17
(a)
(a)
\therefore A \cup B' \ne A' \cup B
Try it yourself 18
(a)
\begin{align}
( A \cup B)'
\end{align}
(b)
\begin{align}
P \cup Q'
\end{align}
Try it yourself 19
(a)
\begin{align}
\xi & = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 \} \\
\\
M & = \{ 2, 3, 5, 7, 11 \} \\
\\
N & = \{ 1, 2, 3, 5, 6, 10 \}
\end{align}
(b)
(c)(i)
\begin{align}
M' \cap N & = \{ 1, 6, 10 \}
\end{align}
(c)(ii)
\begin{align}
M \cup N' & = \{ 2, 3, 5, 7, 11, 4, 8, 9 \}
\end{align}
