Worksheet Ch 4: Operations on Sets

Section A • Bahagian A [12 Marks]

Question 1 [4 Marks]

The universal set $\xi = \{x : 20 \le x \le 35, x \text{ is an integer}\}$.
Set $A = \{x : x \text{ is a multiple of 4}\}$.
Set $B = \{x : x \text{ has digits whose sum is even}\}$.

(a) List all the elements of Set $A$ and Set $B$. [2 marks]
(b) Find $n(A \cap B)'$. [2 marks]

(a) Elements of Set $A$ and Set $B$

$A = \{20, 24, 28, 32\}$

$B = \{20, 22, 24, 26, 28, 31, 33, 35\}$

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(b) $A \cap B = \{20, 24, 28\} \implies n(A \cap B) = 3$. Total elements $n(\xi) = 35 - 20 + 1 = 16$.

$$n(A \cap B)' = 16 - 3 = 13$$

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Question 2 (Venn Diagram Shading) [4 Marks]

On a Venn diagram showing sets $P, Q, R$ within universal set $\xi$, describe the exact regions to shade for:

(a) $(P \cup Q) \cap R'$ [2 marks]
(b) $(P \cap Q)' \cup R$ [2 marks]

(a) Numbering method: Shade elements in $P$ or $Q$ that lie OUTSIDE circle $R$.

Regions inside $(P \cup Q)$ excluding $R$.

(b) Shade everything outside the overlap of $P$ and $Q$, combined with all of circle $R$.

All regions shaded except $(P \cap Q \cap R')$.

Question 3 [4 Marks]

Given that $n(\xi) = 60$, $n(X) = 32$, $n(Y) = 28$, and $n(X \cap Y) = 10$.

(a) Find $n(X \cup Y)$. [2 marks]
(b) Calculate $n(X \cap Y')$. [2 marks]

(a) Formula: $n(X \cup Y) = n(X) + n(Y) - n(X \cap Y)$

$$n(X \cup Y) = 32 + 28 - 10 = 50$$

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(b) Elements in $X$ only: $n(X) - n(X \cap Y)$

$$32 - 10 = 22$$

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Section B • Bahagian B (HOTS / KBAT) [18 Marks]

Question 4 (3-Set School Club Survey) [9 Marks]

A survey of $100$ Form 4 students was conducted to identify club participation in Robotics ($R$), Mathematics ($M$), and Debate ($D$):

$48$ students participate in Robotics ($R$).
$52$ students participate in Mathematics ($M$).
$38$ students participate in Debate ($D$).
$20$ students participate in both Robotics and Mathematics ($R \cap M$).
$15$ students participate in both Robotics and Debate ($R \cap D$).
$18$ students participate in both Mathematics and Debate ($M \cap D$).
$8$ students participate in all three clubs.

(a) How many students participate in only one club? [4 marks]
(b) Calculate the number of students who do not participate in any of these three clubs. [3 marks]
(c) Find the number of students who participate in Robotics or Mathematics but NOT Debate. [2 marks]

(a) Two-club only overlaps:

Only $R \cap M = 20 - 8 = 12$; Only $R \cap D = 15 - 8 = 7$; Only $M \cap D = 18 - 8 = 10$.

Single club only:

Only $R = 48 - (12 + 7 + 8) = 21$

Only $M = 52 - (12 + 10 + 8) = 22$

Only $D = 38 - (7 + 10 + 8) = 13$

$$\text{Only one club} = 21 + 22 + 13 = \mathbf{56}\text{ students}$$

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(b) Total in at least one club: $n(R \cup M \cup D) = 21 + 22 + 13 + 12 + 7 + 10 + 8 = 93$

$$\text{None} = 100 - 93 = \mathbf{7}\text{ students}$$

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$$21 + 22 + 12 = \mathbf{55}\text{ students}$$

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Question 5 (Unknown Elements in Subsets) [9 Marks]

In a group of $75$ tourists, $40$ visited Penang ($P$), $35$ visited Melaka ($M$), and $30$ visited Langkawi ($L$). Given that:

$n(P \cap M) = 16$, $n(P \cap L) = 12$, $n(M \cap L) = 14$

The number of tourists who visited all three destinations is $x$.

$5$ tourists did not visit any of the three destinations.

(a) Form an algebraic equation in terms of $x$ and solve for $x$. [4 marks]
(b) Calculate the number of tourists who visited exactly two destinations. [3 marks]
(c) State the probability that a randomly chosen tourist visited only Penang. [2 marks]

(a) Total in union: $75 - 5 = 70$. Apply 3-set union formula:

$$40 + 35 + 30 - 16 - 12 - 14 + x = 70$$

$$105 - 42 + x = 70 \implies 63 + x = 70$$

$$x = 7\text{ tourists}$$

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(b) Exactly two destinations: $(16-7) + (12-7) + (14-7) = 9 + 5 + 7$

$$=\mathbf{21}\text{ tourists}$$

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$$P(\text{Only } P) = \frac{19}{75}$$

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