Problem-solving Pack

Mathematics

Year 11 · 11.1 Sets and Venn Diagrams

Problem-solving Pack

Name: _________________________________

Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12

Universal set operations. The universal set is

U = \{1, 2, 3, \ldots, 15\}

. Let

A = \{x \in U : x \text{ is a multiple of } 3\}

and

B = \{x \in U : x \text{ is a factor of } 12\}

.

(a) List the elements of

A

B

A \cap B

A \cup B

, and

A'

.
(b) State whether

A

and

B

are mutually exclusive. Justify.
(c) Find

n(A' \cap B)

and

n(A \cup B)'

Working space

2Problem 2 of 12

Venn diagram from sentence. In a class of 30 students, 18 play tennis, 12 play hockey, and 5 play both.

(a) Draw a Venn diagram with all four regions labelled.
(b) How many play neither tennis nor hockey?
(c) Find

P(\text{plays tennis only})

.
(d) Find

n(T \cup H)

and

n((T \cup H)')

Working space

3Problem 3 of 12

Three-set Venn — school subjects. At a school of 100 students: 55 enjoy Maths (M), 48 enjoy Science (S), 30 enjoy Art (A). 22 enjoy M & S, 10 enjoy S & A, 15 enjoy M & A, and 6 enjoy all three.

(a) Use the inclusion–exclusion principle to find

n(M \cup S \cup A)

.
(b) Find the number who enjoy exactly one of the three subjects.
(c) Find the probability that a randomly chosen student enjoys none of the three.

Working space

4Problem 4 of 12

Languages survey. In a class of 25 students, 15 study French, 12 study Spanish, and 4 study neither.

(a) How many study at least one language?
(b) Use the inclusion–exclusion formula to find the number who study both.
(c) Draw a Venn diagram and fill in all four regions.

Working space

5Problem 5 of 12

Sample space — two dice. Two fair six-sided dice are rolled.

(a) State the size of the sample space.
(b) List the outcomes in the event

E

= "the sum is 7".
(c) List the outcomes in the event

F

= "at least one die shows a 6".
(d) Find

n(E \cap F)

and

n(E \cup F)

Working space

6Problem 6 of 12

Three-set Venn — fitness app. A fitness app tracks three habits among 200 users: running (R), cycling (C), swimming (S).

- 80 run, 70 cycle, 60 swim.
- 30 run and cycle, 25 run and swim, 20 cycle and swim.
- 10 do all three.

(a) Find the number who do at least one of the three activities.
(b) Find the number who do none.
(c) Find the number who do exactly two.

Working space

7Problem 7 of 12

Algebraic Venn fill. In a 2-set Venn diagram, "

A

only" contains

2x

elements, "

B

only" contains

x + 5

, "both" contains

x

, and "neither" contains 4 elements. The universal set has 25 elements.

(a) Set up an equation in

x

.
(b) Solve for

x

.
(c) State

n(A)

n(B)

n(A \cap B)

, and

n(A \cup B)

Working space

8Problem 8 of 12

De Morgan in action [EXT]. Let

U = \{1, 2, \ldots, 10\}

A = \{2, 3, 5, 7\}

B = \{2, 4, 6, 8, 10\}

.

(a) Find

A \cup B

and

(A \cup B)'

.
(b) Find

A'

and

B'

and hence find

A' \cap B'

.
(c) Verify your answer to (a) and (b) match De Morgan's law

(A \cup B)' = A' \cap B'

Working space

9Problem 9 of 12

Lifting from a conditional context. In a survey,

n(U) = 50

n(A) = 24

n(B) = 20

n(A \cap B) = 12

.

(a) Find

n(A \cup B)

and

n(A \cup B)'

.
(b) Find

n(A \cap B')

and

n(A' \cap B)

.
(c) Construct a 2-set Venn diagram showing all four regions.

Working space

10Problem 10 of 12

Set-builder & interval. Express each set in interval notation; then describe in words.

(a)

\{x \in \mathbb{R} : -2 \leq x < 5\}

(b)

\{x \in \mathbb{R} : x > 3\}

(c)

\{x \in \mathbb{R} : 0 \leq x \leq 10 \text{ and } x \neq 5\}

Working space

11Problem 11 of 12

Subsets and power set [EXT]. Let

A = \{a, b, c, d\}

.

(a) How many subsets does

A

have?
(b) List the subsets of size 2.
(c) Explain in words why a set with

n

elements has

2^n

subsets.

Working space

12Problem 12 of 12

Modelling — sports club membership. A sports club has 100 members. Each plays at least one of football (F), tennis (T), or swimming (S). 60 play F, 50 play T, 40 play S. 20 play F & T, 15 play F & S, 10 play T & S.

x

members play all three.

(a) Show that

x = 5

.
(b) How many members play exactly one sport?
(c) The treasurer wants to send a discount voucher to members who play more than one sport. How many vouchers are needed?

Working space