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Sets and Venn diagrams

Sets and Venn diagrams

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Statistics

Sets and Venn diagrams

Sampling and bias

Collecting data: types and classes of data

Mean, median, mode and range

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Frequency tables: finding averages

Grouped frequency tables

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Basics of probability

Calculating theoretical probabilities

Probability: Expected and relative frequency

The AND / OR rules

Probability tree diagrams

Conditional probability - Higher

Experimental probability: frequency trees

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Pythagoras' theorem

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Finding percentages and percentage change

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Algebra

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Number

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Multiplying and dividing by powers of 10

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Index laws: negative and fractional indices - Higher

Surds: Simplify, add and subtract - Higher

Rationalising surds - Higher

Standard form calculations

Explainer Video

Tutor: Alice

Summary

Sets and Venn diagrams

​​In a nutshell

A set is a collection of 'things' known as elements. In statistics, sets are usually presented on a Venn diagram. This provides an easy way for the probability of an element, inside a given set, to be worked out.



Set theory

Sets are represented by a list of elements inside two curly brackets: { }


Example 1

The set of even numbers between 111 and 101010:

  • { 2,4,6,82,4,6,82,4,6,8 }
  • {x:x:x: xxx is an even number between one and ten}​

​

Venn diagrams representing sets

On a Venn diagram, each set is represented by a circle which contains all the elements within it.

​

Set Notation

Definition

Venn diagram Representation

n(AAA​)

Number of elements in set AAA.

​The empty set ∅\emptyset∅​​​

A set which has nothing in it.

Universal Set ξ\xiξ or EEE or UUU​​

The set containing all elements.
Maths; Statistics; KS4 Year 10; Sets and Venn diagrams

Complement A′A'A′ or A‾\overline{A}A​​

All elements not in AAA.
Maths; Statistics; KS4 Year 10; Sets and Venn diagrams

Intersection A∩BA \cap BA∩B

All elements in both AAA​ and BBB.
Maths; Statistics; KS4 Year 10; Sets and Venn diagrams

Union A∪BA \cup BA∪B

All elements in either AAA or BBB or both.
Maths; Statistics; KS4 Year 10; Sets and Venn diagrams

Conditional (A given B) A∣BA|BA∣B​​

All elements in AAA​ given they are in BBB.
Maths; Statistics; KS4 Year 10; Sets and Venn diagrams

Disjoint sets

Two sets that share none of the same elements.
Maths; Statistics; KS4 Year 10; Sets and Venn diagrams

​Subset A⊂BA \subset BA⊂B​

​AAA​ is a set inside of BBB.
Maths; Statistics; KS4 Year 10; Sets and Venn diagrams



Finding probabilities from Venn diagrams

You can use Venn diagrams to work out the probability of an event by counting the number of elements in the relevant set and taking that as a fraction out of the total.

​

Example 2

The Venn diagram shows the numbers of students that said they liked green and blue. 

Maths; Statistics; KS4 Year 10; Sets and Venn diagrams

Find the probability that when selected at random, a pupil would like:


1. The colour green:

 N(Year 11 Students)=100\text{N(Year 11 Students)}= 100N(Year 11 Students)=100


Work out the number of students that like green.

n(G)=20+23=43\text{n}(G)= 20+23=43n(G)=20+23=43


Work out the probability that a student likes green.

P(G)=43100 \text{P(}G)= \frac{43}{100}P(G)=10043​


2. Both blue and green:

Work out the number of students that like both blue and green.

n(B∩G)=20\text{n(}B\cap G)= 20n(B∩G)=20


Work out the probability that a student likes both blue and green.

P(B∩G)=20100=15\text{P(}B \cap G)= \frac{20}{100}=\frac{1}{5}P(B∩G)=10020​=51​


3. Blue given they like green:

Take the number of students that like both blue and green as a fraction out of the number that like green.

n(G)=43 n(B∩G)=20P(B∣G)=2043\text{n(}G)=43 \\\text{ n(}B\cap G)= 20 \\\text{P(}B|G)=\frac{20}{43}n(G)=43 n(B∩G)=20P(B∣G)=4320​​​

​​

Note: In the example above 252525 is the number of students that did not fit into either category - they did not like blue or green!

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Listing outcomes

Listing outcomes

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Venn diagrams

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Sets and Venn diagrams

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FAQs - Frequently Asked Questions

What are disjoint sets?

Disjoint sets are two sets that share none of the same elements.

What is the union of two sets?

The union of two sets, A and B, is all the elements in A or B.

What is the intersect of two sets?

The intersect of two sets, A and B is all elements in both A and B.

What does the complement of A mean?

The complement is all elements not in A.

What is the universal set?

The universal set (ξ or E or U) is the set containing all elements.

How do I represent a set?

Sets are represented by a list of elements inside two curly brackets: { }.

What is a set?

A set is a collection of 'things' known as elements.

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