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iGCSE (2021 Edition)

23.01 Set notation and Venn diagrams

Worksheet
Set notation and sample space
1

List all the possible outcomes when a coin is flipped.

2

A standard six-sided die is rolled.

a

List the sample space.

b

List the sample space for rolling a number strictly less than 3.

c

List the sample space for rolling a number divisible by 3.

d

List the sample space for rolling an even number.

3

Uther rolled a standard six-sided die.

a

List all the numbers that the die may land on.

b

Uther rolled a number less than 3. List all the numbers that he could have rolled.

4

List all the elements in the following sets:

a

{\{x:x \text{ is a positive even integers that are less than }16}\}

b

{\{x:x \text{ is an integer between} -8 \text{ and} -3 \text{ (not inclusive)}}\}

c

{\{x:x \text{ is an odd whole number less than } 13}\}

d

\left\{1, \dfrac{1}{10}, \dfrac{1}{100}, \ldots, \dfrac{1}{100\,000}\right\}

e

\left\{2, 4, 8, \ldots, 64\right\}

5

Consider the following sets:

A = \left\{1, 2, 4, 8, 16\right\}, \quad B = \left\{1, 2, 4, 8\right\}

List all the elements in:

a
A \text{ or } B
b
Both A and B
Set notation (Extended)
6

Consider the set A = \left\{2, 4, 6, 8\right\}. Construct set builder notation for A.

7

Consider the following sets:

A = \left\{8, 12, 16, 20, 24, 28\right\}, B = \left\{8, 12, 16, 28\right\}, C = \left\{12, 16\right\}, \text{and } D = \left\{4, 8, 12, 16, 20, 24, 28\right\}

Determine whether the following statements are true or false:

a

B \subset D

b

C\subset B

c

A \subset B

d

\emptyset \subset D

8

Write \subseteq or \nsubseteq to make each of the following statements true:

a

\left\{1, 2, 4\right\} ⬚ \left\{1, 2, 4, 7, 8\right\}

b

\left\{8, 9\right\} ⬚ \left\{3, 4, 7, 8, 9\right\}

c

\left\{3, 4, 9\right\} ⬚ \left\{2, 4, 7, 8, 9\right\}

d

\left\{1, 3, 8\right\} ⬚ \left\{1, 4, 7, 8\right\}

9

Write \in or \notin to make each of the following statements true:

a

5 \ ⬚ \left\{2, 5, 6, 9\right\}

b

11\ ⬚\left\{4, 8, 12, 14\right\}

10

Consider the following sets:

  • A= \{ \text{letters of the word `parallelogram'}\}
  • B= \{ \text{letters of the word `gallop'}\}
  • C= \{ \text{letters of the word `llama'}\}
a

State the set of elements contained in both A and B.

b

Is B a proper subset of A?

c

State the set of elements contained in A or C.

11

Explain the meaning of "the complement of set A."

12

A set is defined as A=\left\{ x:0\leq x \leq 5 \text{ and }x \text{ is an integer} \right\}.

a

List the elements of set A.

b

Is set B=\left\{2, 3, 5\right\} a subset of set A?

13

Consider the sets A=\left\{ x:-6\lt x \lt 6 \text{ and }x \text{ is an integer} \right\} and \\ B=\left\{ x:-1\leq x \leq 10 \text{ and }x \text{ is an integer} \right\}.

a

Find n(A).

b

Find n(B).

c

Find n(A \cap B).

d

Find n(A \cup B).

14

State whether the following points are in the set A=\left\{(x, y) : y = 5x-7 \right\}:

a
\left(9,40\right)
b
\left(2,3\right)
c
\left(0,7\right)
d
\left(\dfrac{3}{2},\dfrac{1}{2}\right)
15

Consider the sets A=\left\{(x, y) : y = 3x-4 \right\} and B=\left\{(x, y) : y = 2x+6 \right\}.

a

Explain why n(A \cap B)=1.

b

Find A \cap B.

Operations on sets
16

Explain the meaning of the following statements:

a

Union of set A and set B.

b

Intersection of set A and set B.

17

P and Q are sets of vegetable types:

P= \{carrots, cauliflowers, beans\}; Q = \{cauliflowers, potatoes\}

There are no other vegetable types in universal set U.

a

Is P \cup Q the set of all vegetable types?

b

List the elements in the set P \cap Q.

18

Suppose set A = \left\{3, 4, 5, 6, 7\right\} and set B = \left\{3, 7, 8, 9\right\}. Find A \cap B.

19

List the elements of A \cap B given the following sets:

a

A= \{ \text{even numbers} \} and B = \{ \text{square numbers less than}\text{ } 100 \}.

b

A= \{ \text{multiples of} \text{ } 5\} and B= \{ \text{positive numbers less than} \text{ } 50 \}

20

If A is the set of factors of 24, and B is the set of factors of 36, then list the elements of:

a

B \cup A

b

A \cap B

21

Set A is the set of possible outcomes from rolling a standard die, and set B is the set of possible outcomes from rolling an eight-sided die. List the elements of the following sets:

a

A

b

B

c

A \cap B

d

A \cup B

Complement of sets (Extended)
22

P and Q are sets of flower varieties:

P= \{roses, lillies, daisies\}; Q= \{lillies, sunflowers\}

There are no other flower varieties in universal set U.

a

List the elements in the following sets:

i
Q'
ii
Q \cap P
iii
Q \cup P
iv
Q' \cap P
b

Is \left(P \cup Q\right) \rq empty?

23

The sets V = \left\{21, 8, 30, 9, 28\right\} and W = \left\{21, 8, 30, 9, 28, 7, 13, 12, 26\right\} are such that there are no other elements outside of these two sets.

a

Is V a proper subset of W?

b

State n\left(V\right), the order of V.

c

List the elements of V'.

d

List the elements of the universal set.

e

Find W'.

24

Consider the following sets:

A = \left\{1, 2, 3, 4, 5, 6, 7\right\},B = \left\{1, 2, 3, 4\right\}

If there are no elements contained outside of these sets, find:

a
B'
b
A'
25

Consider the following sets:

\\

A = \{ \text{people who like football} \}

B = \{ \text{people who like softball} \}

C = \{ \text{people who like swimming} \}

D = \{ \text{people who do not like any of these} \}

a

Describe set B'.

b

Describe set D'.

26

Consider the sets A=\left\{ x:-7\leq x \leq 10 \text{ and }x \text{ is an integer} \right\}, B=\left\{-2, 4, 8\right\} and \\ C=\left\{3, -5, 11\right\}.

a

Is set B a subset of set A?

b

Is set C a subset of set A?

c

List the elements in the set A \cap C.

d

List the elements in the set A' \cap C.

e

Find n \left( A \cup C \right).

f

Find n \left( A \cap C' \right).

Venn diagrams
27

For each of the following, identify the Venn diagram that best represents the sets A and B:

Venn diagram 1

Venn diagram 3

Venn diagram 2

Venn diagram 4

a

A = \left\{\text{Earth}\right\}, B = \left\{\text{Planets}\right\}

b

A = \left\{6, 8, 4, 2, 7\right\}, B = \left\{3, 6, 1\right\}

c

A = \left\{\text{w}, \text{i}, \text{n}, \text{d}\right\}, B = \left\{\text{e}, \text{a}, \text{r}, \text{t}, \text{h}\right\}

d

A = \left\{\text{Animals found in Australia}\right\}, B = \left\{\text{Animals found in NSW}\right\}

28

We are interested in the colour of a card randomly drawn from a standard deck. Draw a Venn diagram to illustrates this.

29

Consider the Venn diagram:

List the elements in:

a

A \cap B

b

A \cup B

Venn diagrams (Extended)
30

Consider the Venn diagram:

List the elements in:

a

A \rq

b

B \rq

31

Consider the Venn diagram:

List the elements in:

a

A

b

U

c

B \rq

32

Consider the Venn diagram:

List the elements in:

a

A \cap B \rq

b

\left(A \cup B\right) \rq

33

Consider the following Venn diagram:

a

Is A a subset of B?

b

Is A a proper subset of B?

34

Consider the following diagram:

a

Find the set A \cap C.

b

Find the set \left(B \cap C\right) '.

c

Find the set A \cap B \cap C.

d

Find the set A \cap \left( B \cup C \right).

e

Find the set \left(A \cap B \right)\rq.

35

Consider the following Venn diagram:

a

Is (A \cap B)' equal to A'\cup B' for all sets?

b

Is A' \cap B' equal to A \cup B' for all sets?

36

Consider the following Venn diagram:

Find the elements in the following:

a

A \cap B' \cap C'

b

A \cap B \cap C'

c

A' \cap B \cap C'

d

A \cap B' \cap C

e

A \cap B \cap C

f

A' \cap B \cap C

g

A' \cap B' \cap C

h

A' \cap B' \cap C'

37

The Venn diagram shows the number of students in a school playing Rugby League (L), Rugby Union (U), both or neither.

a

Complete the table of values below:

Play Rugby LeagueDon't play Rugby League
Play Rugby Union
Don't play Rugby Union
b

Find the following:

i

n(L\cap{U}')

ii

n(L)

iii

n(U)

iv

n(U\cap L')

v

n({L}')

vi

n({U}')

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Outcomes

0580C1.2A

Understand notation of Venn diagrams.

0580C1.2B

Definition of sets e.g. A = {x: x is a natural number} B = {a, b, c, …}.

0580E1.2A

Use language, notation and Venn diagrams to describe sets and represent relationships between sets.

0580E1.2B

Definition of sets e.g. A = {x: x is a natural number} B = {(x, y): y = mx + c} C = {x: a ⩽ x ⩽ b} D = {a, b, c, …}.

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