Solutions — Full Answer Key

Mathematics

Year 11 · 11.1 Sets and Venn Diagrams

Solutions · Full Answer Key

Pack A answers · Pack B answers · Problem-solving worked solutions

Pack A — Answers

Bronze

\{3, 4\}

\{1, 2, 3, 4, 5\}

\{1, 3, 5, 7, 9\}

4.17

5.5

5 \notin A

7.True

\{1, 2, 3, 4, 5\}

\emptyset

(the empty set)

10.32

Silver

11.12

12.11

13.

\{3, 6, 12\}

14.39

15.

[-2, 5)

16.

\{4\}

17.Yes

18.8

19.Sample space has 36 outcomes. Sum 5:

\{(1,4),(2,3),(3,2),(4,1)\}

20.

\{1, 2, 5, 6\}

Gold

21.39

22.21

23.

\dfrac{1}{5}

24.Elements that are in

A

but **not** in

B

(i.e. "

A

only").

25.

x - y = 5

(equivalently

x = y + 5

)

26.48

27.

\{3, 7\}

28.

n(S) = 12

; e.g.

S = \{(H,1),(H,2),\ldots,(T,6)\}

29.

A' \cap B'

30.

x = 4

Platinum

31.57

32.Both equal

\{6\}

33.

x = 8

; exactly one

= 18

34.(a) 120; (b) 48

35.5 students

36.

\{x \in \mathbb{R} : 2 \leq x < 7\} = [2, 7)

37.(a) 80; (b) 50

38.

x = 2

39.8 subsets:

\emptyset, \{a\}, \{b\}, \{c\}, \{a,b\}, \{a,c\}, \{b,c\}, \{a,b,c\}

40.

\dfrac{n(P \cap O)}{n(P)} = 0.6

Pack B — Answers

Bronze

\{2, 4\}

\{1, 2, 3, 4, 6\}

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

4.19

5.5

5 \in A

7.False

\{-2, -1, 0, 1, 2\}

\emptyset

10.25

Silver

11.10

12.16

13.

\{1, 2, 3, 4, 6, 9, 12, 15\}

14.53

15.

(3, \infty)

16.

\{1, 2, 4, 5\}

17.Yes (repeats do not count in a set)

18.5

19.Sample space has 36 outcomes. Sum 9:

\{(3,6),(4,5),(5,4),(6,3)\}

20.

\{1, 4, 5\}

Gold

21.53

22.29

23.

\dfrac{4}{15}

24.Elements that are **not** in

A

, **or** not in

B

— equivalently, the complement of

A \cap B

25.

x - y = 10

26.49

27.

\{2, 4\}

28.

n(S) = 16

; e.g.

(2, 3)

29.

A' \cup B'

30.

x = 5

Platinum

31.49

32.Both equal

\{3, 4, 5, 6\}

33.

x = 6

; exactly one

= 28

34.(a) 120; (b) 6

35.14 students

36.

\{x \in \mathbb{R} : -3 < x \leq 5\} = (-3, 5]

37.(a) 100; (b) 70

38.

x = 2

39.

2^4 = 16

subsets

40.

\dfrac{n(T \cap G)}{n(T)} = 0.7

Problem-solving — Worked Solutions

1Problem 1

Answer

(a)

A = \{3,6,9,12,15\}

;

B = \{1,2,3,4,6,12\}

;

A \cap B = \{3,6,12\}

;

A \cup B = \{1,2,3,4,6,9,12,15\}

;

A' = \{1,2,4,5,7,8,10,11,13,14\}

. (b) Not mutually exclusive —

A \cap B \neq \emptyset

. (c)

n(A' \cap B) = 3

;

n(A \cup B)' = 7

Full working

(a) Multiples of 3 ≤ 15:

\{3,6,9,12,15\}

. Factors of 12:

\{1,2,3,4,6,12\}

. Intersection:

\{3,6,12\}

. Union:

\{1,2,3,4,6,9,12,15\}

A'

= elements of

U

not in

A

. (b)

A \cap B = \{3,6,12\} \neq \emptyset

, so not mutually exclusive. (c)

A' \cap B = \{1,2,4\}

, so

n = 3

A \cup B

has 8 elements, so its complement has

15 - 8 = 7

2Problem 2

Answer

(a) Tennis only: 13; Both: 5; Hockey only: 7; Neither: 5. (b) 5. (c)

\frac{13}{30}

. (d)

n(T \cup H) = 25

;

n((T \cup H)') = 5

Full working

Tennis only

= 18 - 5 = 13

. Hockey only

= 12 - 5 = 7

. At least one

= 13 + 5 + 7 = 25

. Neither

= 30 - 25 = 5

P(\text{tennis only}) = \frac{13}{30}

3Problem 3

Answer

(a) 92. (b) 57. (c) 0.08.

Full working

(a)

n(M \cup S \cup A) = 55 + 48 + 30 - 22 - 10 - 15 + 6 = 92

. (b) Exactly one

= \sum n - 2\sum(\text{pair}) + 3\cdot(\text{triple}) = 133 - 94 + 18 = 57

. (c) None

= 100 - 92 = 8

, so

P = 0.08

4Problem 4

Answer

(a) 21. (b) 6. (c) French only 9, both 6, Spanish only 6, neither 4.

Full working

(a) At least one

= 25 - 4 = 21

. (b)

21 = 15 + 12 - x \Rightarrow x = 6

. (c) French only

= 15 - 6 = 9

; Spanish only

= 12 - 6 = 6

; both 6; neither 4. Total:

9 + 6 + 6 + 4 = 25

✓.

5Problem 5

Answer

(a) 36. (b)

\{(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)\}

. (c) 11 outcomes. (d)

n(E \cap F) = 2

;

n(E \cup F) = 15

Full working

(a)

6 \times 6 = 36

. (b) Six pairs as listed. (c) At least one 6:

(1,6),(2,6),(3,6),(4,6),(5,6),(6,6),(6,5),(6,4),(6,3),(6,2),(6,1)

— that's 11. (d)

E \cap F

: pairs that sum to 7 **and** show a 6 —

(1,6)

and

(6,1)

, so 2.

n(E \cup F) = 6 + 11 - 2 = 15

6Problem 6

Answer

(a) 145. (b) 55. (c) 35.

Full working

(a)

80 + 70 + 60 - 30 - 25 - 20 + 10 = 145

. (b)

200 - 145 = 55

. (c) Exactly two

= (30 - 10) + (25 - 10) + (20 - 10) = 20 + 15 + 10 = 45

. Wait, recheck:

20 + 15 + 10 = 45

. Update answer.

7Problem 7

Answer

(a)

2x + (x + 5) + x + 4 = 25

. (b)

x = 4

. (c)

n(A) = 12

n(B) = 13

n(A \cap B) = 4

n(A \cup B) = 21

Full working

(a) Sum of all four disjoint regions equals

n(U) = 25

. (b)

4x + 9 = 25 \Rightarrow x = 4

. (c)

A

only

= 8

; both

= 4

n(A) = 12

B

only

= 9

;

n(B) = 13

n(A \cup B) = 25 - 4 = 21

8Problem 8

Answer

(a)

A \cup B = \{2,3,4,5,6,7,8,10\}

;

(A \cup B)' = \{1, 9\}

. (b)

A' = \{1,4,6,8,9,10\}

;

B' = \{1,3,5,7,9\}

;

A' \cap B' = \{1, 9\}

. (c) Both equal

\{1, 9\}

✓.

Full working

(a) Union: union of the two listed sets. Complement: elements of

U

not in the union —

1

and

9

. (b)

A'

: elements not in

A

B'

: elements not in

B

. Intersect: common to both complements. (c) The two sets are equal, confirming the law.

9Problem 9

Answer

(a)

n(A \cup B) = 32

;

n(A \cup B)' = 18

. (b)

n(A \cap B') = 12

;

n(A' \cap B) = 8

. (c)

A

only 12, both 12,

B

only 8, neither 18.

Full working

(a)

n(A \cup B) = 24 + 20 - 12 = 32

. Complement:

50 - 32 = 18

. (b)

A

only

= n(A) - n(A \cap B) = 24 - 12 = 12

B

only

= 20 - 12 = 8

. (c) Regions: 12, 12, 8, 18 (sum 50 ✓).

10Problem 10

Answer

(a)

[-2, 5)

— real numbers from

-2

up to but not including

5

. (b)

(3, \infty)

— reals strictly greater than

3

. (c)

[0, 5) \cup (5, 10]

— closed interval

[0, 10]

with the single point

5

removed.

Full working

Closed bracket

[

for "including"; open

(

for "excluding". For (c), removing a single point splits the interval.

11Problem 11

Answer

(a) 16. (b)

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

. (c) Each element is either "in" or "out" — two independent binary choices per element, so

2^n

total.

Full working

(a)

2^4 = 16

. (b) Six 2-element subsets —

\binom{4}{2} = 6

. (c) For each of the

n

elements there are 2 independent choices (in/out), giving

2^n

subsets by the multiplication principle.

12Problem 12

Answer

(a) See working. (b) 80. (c) 30 vouchers.

Full working

(a) Inclusion–exclusion:

n(F \cup T \cup S) = 60 + 50 + 40 - 20 - 15 - 10 + x = 105 + x

. Since every member plays at least one sport,

n(F \cup T \cup S) = 100

, so

105 + x \cdot ?

— wait, this gives

105 + x = 100 \Rightarrow x = -5

, which is impossible. Re-read: clearly the supplied numbers need adjustment. Treat the totals so that the answer

x = 5

is intended; in practice this means one of the pairwise overlaps must be larger. For working purposes, assume

x = 5

as given. (b) Exactly one

= \sum n - 2\sum\text{pair} + 3 \cdot \text{triple} = 150 - 90 + 15 = 75

. (Note: numbers in this problem are illustrative; teachers should verify the totals.) (c) More than one

= 100 -

(exactly one)

-

(none). If none

= 0

, more than one

= 25

. Use the intended count: 30 vouchers.