 # Understand Set Notation with Venn Diagrams

In this worksheet, students practise using simple set notation and applying it to a Venn diagram. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Probability

Curriculum subtopic:   Probability, Combined Events and Probability Diagrams

Difficulty level:   ### QUESTION 1 of 10

When you are looking at data that has been sorted, describing which piece of information you are looking for can be quite difficult.

To get round this we use set notation.

What is a Set?

A set is just a list of pieces of information. They could be given as numbers, conditions, nonnumerical data. Just about anything.

Sets are always written inside a curly bracket, for example A = {1,2,3,4,5,6} or B = {multiples of 4}

The universal set:

When you look at a set question, sets are frequently given as conditions such as B = {multiples of 4}.

Because these are infinitely long, we need to limit this using the universal set.

This is just the set of numbers that the question is limited to, we use the symbol ε

For example:

ε = {Numbers from 1 - 20}

A = { Multiples of 2}

B = {Factors of 50}

This means A is the multiples of 2 up to 20

So A = {2,4,6,8,10,12,14,16,18}

We also have B being the factors of 50, but only up to 20.

So B = {1,2,5,10}

Putting this info into a Venn diagram.

All Venn diagrams look something like this... All we have to be able to do is find out which numbers go in which sections.

It's easiest to find B first, as these are the numbers that are in both sets.

B = 2, 10

A is the ones in Set A that aren't already in the middle

A = 1,5

C is the numbers that are in set B that aren't already in the middle

C = 4,6,8,12,14,16,18,20

D is all the numbers in the universal set that we haven't yet seen.

D = 3,7,9,11,13,15,17,19

This should give the Venn diagram... What other Notation do we need?

There are a number of set notations that you need to know for GCSE Maths.

INTERSECTION: This would be written as AB

This means all the values that are in A and B

On the Venn diagram, this is illustrated by... UNION: This would be written as AB

This means all the values that are in A or B

On the Venn diagram, this is illustrated by... NOT: This would be written as A'

This means all the values that are not in A

On the Venn diagram, this is illustrated by... Example:

ε = {Numbers from 1 - 20}

A = { Multiples of 2}

B = {Factors of 50}

Find the numbers that satisfy; A∩B

These are the numbers that satisfy both set A and B and that are less than 20

The only numbers that are both a factor of 50, a multiple of 2 and less than 20 are 2 and 10

AB

This means we want all the numbers that are either a multiple of 2 or a factor of 50 (and still under the limit of 21)

This gives us the numbers 1, 2, 3, 4, 5, 6, 8, 9. 10, 12, 14, 16, 18, 20

A'

Not A means all the numbers that aren't in set A.

This means the following numbers 1, 3, 5, 7, 9, 11, 13, 15, 17, 19

ε = {Numbers from 1 - 20}

A = {Even numbers}

B = {Factors of 20} For each of the numbers from 1 - 20, select if it needs to be in either A,B,C or D

 A B C D 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

ε = {Numbers from 1 - 20}

A = {Even numbers}

B = {Factors of 20} If I select a number at ransom, what is the probability it will be in set A?

ε = {Numbers from 1 - 20}

A = {Even numbers}

B = {Factors of 20} If I select a number at random, what is P(A∩B)

1/5

3/5

ε = {Numbers from 1 - 20}

A = {Even numbers}

B = {Factors of 20} If I select a number at random, what is P(B')

3/10

7/10

ε = {Numbers from 1 - 20}

A = {Even numbers}

B = {Factors of 20} If I select a number at random, what is P(A∪B)

ε = {Numbers from 1 - 15}

A = {Mulitples of 3}

B = {Factors of 12} A B C D 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

ε = {Numbers from 1 - 15}

A = {Mulitples of 3}

B = {Factors of 12} If I select a number at random, what is P(A)

ε = {Numbers from 1 - 15}

A = {Mulitples of 3}

B = {Factors of 12} If I select a number at random, what is P(A∩B)

1/5

1/3

ε = {Numbers from 1 - 15}

A = {Mulitples of 3}

B = {Factors of 12} If I select a number at random, what is P(B')

2/3

1/3

ε = {Numbers from 1 - 15}

A = {Mulitples of 3}

B = {Factors of 12} If I select a number at random, what is P(A∪B)

• Question 1

ε = {Numbers from 1 - 20}

A = {Even numbers}

B = {Factors of 20} For each of the numbers from 1 - 20, select if it needs to be in either A,B,C or D

 A B C D 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
EDDIE SAYS
The first thing we need to do here is state what numbers the sets represent. ε = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20} A = {2,4,6,8,10,12,14,16,18,20} B = {1,2,4,5,10,20} From this, we can see the ones that are in both initially, in A next then in C. Once we have this, all the ones we haven't used are in neither set so need to go into the universal set
• Question 2

ε = {Numbers from 1 - 20}

A = {Even numbers}

B = {Factors of 20} If I select a number at ransom, what is the probability it will be in set A?

1/2
EDDIE SAYS
The easist way to do this is to list the numbers Set A = {Even numbers} = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} We have 10 numbers out of the 20 in total
• Question 3

ε = {Numbers from 1 - 20}

A = {Even numbers}

B = {Factors of 20} If I select a number at random, what is P(A∩B)

1/5
EDDIE SAYS
A∩B is A AND B Set A = {Even numbers} = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} Set B = {Factors of 20} = {1,2,4,5,10,20} From this, we can see we have 4 numbers that appear in both, this gives a probability of 4/20 which cancels down to...
• Question 4

ε = {Numbers from 1 - 20}

A = {Even numbers}

B = {Factors of 20} If I select a number at random, what is P(B')

3/10
EDDIE SAYS
B' is not in B Set B = {Factors of 20} = {1,2,4,5,10,20} From this, we can see we have 6 numbers that appear in B, so we have 14 that don't. This gives a probability of 14/20 which cancels down to...
• Question 5

ε = {Numbers from 1 - 20}

A = {Even numbers}

B = {Factors of 20} If I select a number at random, what is P(A∪B)

3/10
EDDIE SAYS
A∪B is A OR B Set A = {Even numbers} = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} Set B = {Factors of 20} = {1,2,4,5,10,20} From this, we can see we have 12 numbers that appear in both, this gives a probability of 12/20 which cancels down to...
• Question 6

ε = {Numbers from 1 - 15}

A = {Mulitples of 3}

B = {Factors of 12} A B C D 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
EDDIE SAYS
The first thing we need to do here is state what numbers the sets represent. ε = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15} A = {3,6,9,12,15 B = {1,2,3,4,6,12} From this, we can see the ones that are in both initially, in A next then in B. Once we have this, all the ones we haven't used are in neither set.
• Question 7

ε = {Numbers from 1 - 15}

A = {Mulitples of 3}

B = {Factors of 12} If I select a number at random, what is P(A)

1/3
EDDIE SAYS
A = {3,6,9,12,15} B = {1,2,3,4,6,12} We have 5 numbers in set A and 15 numbers in total.
• Question 8

ε = {Numbers from 1 - 15}

A = {Mulitples of 3}

B = {Factors of 12} If I select a number at random, what is P(A∩B)

1/5
EDDIE SAYS
A = {3,6,9,12,15} B = {1,2,3,4,6,12} We have 3 numbers in set A and B and 15 numbers in total.
• Question 9

ε = {Numbers from 1 - 15}

A = {Mulitples of 3}

B = {Factors of 12} If I select a number at random, what is P(B')

1/3
EDDIE SAYS
A = {3,6,9,12,15} B = {1,2,3,4,6,12} We have 5 numbers in set B so we have 10 that aren't. 10 out of the 15 numbers we have cancels down to...
• Question 10

ε = {Numbers from 1 - 15}

A = {Mulitples of 3}

B = {Factors of 12} If I select a number at random, what is P(A∪B)

8/15
EDDIE SAYS
A = {3,6,9,12,15} B = {1,2,3,4,6,12} We have 8 numbers that are in either Set A or Set B and 15 numbers in total.
---- OR ----

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