Set Notation & Venn Diagrams

There are 31 students in a class.

The only languages available for the class to study are French and Spanish.

17 students study French.

15 students study Spanish.

6 students study neither French not Spanish.

Using a Venn diagram, or otherwise, work out how many students study only one language.

A garage tests cars for faults. 

There are three types of fault – braking, steering and lighting.

A car fails the test if it has one or more of these three types of fault.

Last week, 11 cars had breaking faults, 

                   9 cars had steering faults,

                   7 cars had lighting faults, 

                   no car had both steering faults and lighting faults, 

                   2 cars had both braking faults and steering faults, 

                   3 cars had both braking faults and lighting faults.

By drawing a Venn diagram, or otherwise, find the number of cars which failed the test last week.

ξ = {positive whole numbers less than 13}

A = {even numbers}

B = {multiples of 3}

C = {prime numbers}

List the members of the set

Is it true that 14 ∈ A?

Tick (✔) the appropriate box.

◻ Yes         ◻ No

Explain your answer.

The Venn diagram shows a universal set ξ and three sets A, B and C.

2, 4, 7, 3, 6 and 10 represent the number of elements in each set.

Find

n(A ∩ B)

n(B′)

n(A ∩ C′)

n(B′ ∩ C′)

The Venn diagram shows all of the elements in sets A, B and ξ.

Write down the elements in A′, 

Find n(A ∩ B)′,

Find the elements in (A ∩ B) ∪ (A ∪ B)′,

A ∩ C = ∅

B ∪ C = {5, 6, 7, 8, 9}

n(C) = 3

Write down the elements in C.

On the Venn diagram, shade the set (A ∪ B) ∩ C′.

Use set notation to describe the shaded region in the Venn diagram below.

ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}

A = {3, 7, 11, 13}

B = {3, 6, 9, 12, 13}

C = {2, 3, 5, 6, 7, 8}

Complete the Venn diagram

List the members of the set B′ ∩ C.

List the members of the set (A ∪ C)′ ∩ B.

Find n(A′∩ B′)

Each student in a group of 32 students was asked the following question.

“Do you have a desktop computer (D), a laptop (L) or a tablet (T)?”

Their answers showed that,

19 students have a desktop computer,

17 students have a laptop,

16 students have a tablet,

9 students have both a desktop computer and a laptop,

11 students have both a desktop computer and a tablet,

7 students have both a laptop and a tablet,

5 students have all three.

Using this information, complete the Venn diagram to show the number of students in each appropriate subset.

Some students were asked the following question.

“Which of the subjects Russian (R), French (F) and German (G) do you study?”

Of these students

4 study all three of Russian, French and German
10 study Russian and French
13 study French and German
6 study Russian and German
24 study German
11 study none of the three subjects
the number who study Russian only is twice the number who study French only.

Let be the number of students who study French only.

Show all this information on the Venn diagram, giving the number of students in each appropriate subset, in terms of where necessary.

Given that the number of students who were asked the question was 80, work out the number of these students that study Russian.

 = {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}

= {5, 10, 15}

= {7, 8, 9, 11, 12, 13, 14}

= {4, 6, 7, 8, 14}

Complete the Venn diagram for this information.

Some students in a school were asked the following question.

“Do you have a dog (), a cat () or a rabbit ()?”

Of these students

   28 have a dog
   18 have a cat
   20 have a rabbit
    8 have both a cat and a rabbit
    9 have both a dog and a rabbit
     have both a dog and a cat
    6 have a dog, a cat and a rabbit
    5 have not got a dog or a cat or a rabbit

Using this information, complete the Venn diagram to show the number of students in each appropriate subset.
Give the numbers in terms of  where necessary.

Given that a total of 50 students answered the question,

work out the value of .

= ........................

Find 

ξ = {Students in year 12}

G = {Students who study German}

F = {Students who study French}

M = {Students who study Maths}

G ∩ M = ∅

Use this information to write a statement about the students who study German in Year 12.

Preety is a student in Year 12,

Preety ∉ F.

Use this information to write a statement about Preety.

A = {2, 4, 6, 8, 10}

A ∩ B = {2, 4} 

A ∪ B = {1, 2, 3, 4, 6, 8, 10}

List all the members of set B.

A = {s, u, p, e, r}

B = {c, o, m, p, u, t, e, r}

List the members of the set

X = {prime numbers}

Y = {factors of 12}

Is it true that X ∩ Y = ∅?

Tick (✔) the appropriate box.

◻  Yes           ◻ No

Explain your answer.

ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

A = {odd numbers}

P = {prime number}

List the members of the set A ∩ P.

A, B and C are three sets.

A ∩ B = ∅ and C ⊂ A

Complete the Venn diagram to show the sets B and C.

On the Venn diagram, shade the region that represents A ∩ C’.

The Venn diagram shows a universal set ξ and three sets A, B and C.

7, 6, 3, 2 and 10 represent the number of elements in each set.

Find

n(A ∪ B)

n(A’)

n(B ∩ C’)

n(A’ ∪ B’)

ξ = {whole numbers}

A = {factors of 100}

B = {multiples of 5}

List the members of the set A ∩ B.

ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9}

A = {1, 3, 5, 7}

B = {2, 4, 6, 8}

Explain why A ∩ B = ∅.

x ∈ ξ and x ∉ A ∪ B.

Write down the value of x.

x = ……………………………………… 

A ∩ C = {3, 7}, B ∩ C = {8} and A ∪ B ∪ C = ξ.

List all the members of C.

ξ = {positive whole numbers less than 19}

A = {odd numbers}

B = {multiples of 5}

C = {multiples of 4}

List the members of the set

D = {prime numbers}

Is it true that C ∩ D = ∅?

Tick (✔) the appropriate box.

◻ Yes       ◻ No

Explain your answer.

ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {even numbers}

B = {multiples of 3}

List the members of set B’.

Find n(A’ ∩ B).

x is a member of ξ,

x ∈ B, 

x ∉ A.

What are the possible values of x?

A and B are two sets.

n(ξ) = 36

n(B) = 21

n(A ∩ B) = 8 

n(A’) = 18

Complete the Venn diagram to show the number of elements in each region.

Find

The Venn diagram shows a universal set ξ and three sets X, Y and Z.

The numbers shown represent the numbers of elements.

n(X’) = 14

n(Z) = 14

Complete the Venn diagram.

Find

The Venn diagram shows a universal set and three sets and .

6, 3, 8, 2, 5 and 4 represent the numbers of elements.

Find

The Venn diagram shows a universal set, and sets and

12, 5, 9, 10, 6, 3, 4 and 8 represent the numbers of elements.

Find