(a) The Venn diagram shows set \(X\) and set \(Y\).
(i) List the elements of \(X\).
(ii) Find \(n(Y’)\).
(b) In each Venn diagram, shade the required region:
(1) \(P \cup Q\)
(2) \(P \cap Q\)
(c) \(\% = \{positive integers < 13\}\)
\[ A = \{x : x < 9\}, \quad B = \{x : x \text{ is even}\}, \quad C = \{x : x \text{ is a multiple of 3}\} \]
(i) Complete the Venn diagram.
(ii) Find \(n(A’ \cup (B \cap C))\).
▶️ Answer/Explanation
(a)(i) r, l, t, e, a
Looking at the Venn diagram, the elements inside circle X are: r, l, t, e, a.
(a)(ii) 2
\(Y’\) means “not in Y”. From the diagram, elements outside Y are “r” and “a”, so \(n(Y’) = 2\).
(b) Venn Diagram Shading:
(1) For \(P \cup Q\): Shade both entire circles P and Q completely.
(2) For \(P \cap Q\): Shade only the overlapping region where both circles intersect.
(c)(i) Venn Diagram Completion:
Fill the Venn diagram as follows:
- Only A: 1,2,4,5,7,8
- A∩B: 2,4,6,8
- A∩C: 3,6
- B∩C: 12
- A∩B∩C: 6
- Only B: 10
- Only C: 9
- Outside: 11
(c)(ii) 5
Steps:
- \(A’ = \{9,10,11,12\}\) (numbers ≥9)
- \(B \cap C = \{6,12\}\) (even multiples of 3)
- Union gives \(\{6,9,10,11,12\}\) → 5 elements
(a) In the Venn diagram, shade the region \( P’ \cup Q \).
(b) There are 50 students in a group.
34 have a mobile phone (\( M \)).
39 have a computer (\( C \)).
5 have no mobile phone and no computer.
Complete the Venn diagram to show this information.
(c) The Venn diagram shows the number of students in a group of 30 who have brothers (B), sisters (S) or cousins (C).
(i) Write down the number of students who have brothers.
(ii) Write down the number of students who have cousins but do not have sisters.
(iii) Find \( n(B \cup S \cup C)’ \).
(iv) Use set notation to describe the set of students who have both cousins and sisters but do not have brothers.
(v) One student is picked at random from the 30 students. Find the probability that this student has cousins.
(vi) Two students are picked at random from the students who have cousins. Calculate the probability that both these students have brothers.
(vii) One student is picked at random from the 30 students.
Event A: This student has sisters.
Event B: This student has cousins but does not have brothers.
Explain why event A and event B are equally likely.
▶️ Answer/Explanation
(a) 
(b) 
(c)(i) 8 students have brothers (sum of numbers in B circle).
(c)(ii) 11 students have cousins but no sisters (number in C only region).
(c)(iii) 2 students have none of B, S, or C (number outside all circles).
(c)(iv) \( C \cap S \cap B’ \) describes students with cousins and sisters but no brothers.
(c)(v) Probability is \( \frac{19}{30} \) (total in C circle divided by total students).
(c)(vi) Probability is \( \frac{2}{57} \) (4/19 × 3/18 for picking two with brothers from cousins group).
(c)(vii) Both events have 15 students, making them equally likely with probability \( \frac{15}{30} \).