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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

Simple charts and graphs

Pie charts

Scatter graphs

Frequency tables: finding averages

Grouped frequency tables

Box plots - Higher

Cumulative frequency - Higher

Histograms and frequency density - Higher

Interpreting data

Comparing data sets

Probability

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

Trigonometry

Pythagoras' theorem

Sin, cos, tan

Trigonometry: Finding angles and sides

Exact trigonometric values

Sine and cosine rules - Higher

3D Pythagoras - Higher

3D Trigonometry - Higher

Vectors

Vectors - Higher

Angles and geometry

Angles: types, notation and measuring

Basic angle rules

Angles in parallel lines

Circle theorems - Higher

Constructing triangles: SSS, SAS, ASA

Construction: angle and perpendicular bisectors

Construction: Loci

Bearings

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Shapes and area

Properties of 2D shapes

Congruence: conditions for congruent triangles

Similar shapes: Scaling

The four transformations

Area and perimeter: Formulae

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3D shapes: faces, edges, vertices

Surface area of 3D shapes: Nets, formulae

Volume of 3D shapes: Formulae

Volume of 3D shapes: Comparing, rates of flow

Area and volume scale factors

Projections and elevations of 3D shapes

Ratio proportion and rates of change

Ratio

Direct and inverse proportion

Finding percentages and percentage change

Compound growth and decay

Converting units: metric and imperial

Converting units: area and volume

Time intervals: converting units of time

Speed, density and pressure: Formulae and units

Graphs

Coordinates and midpoints

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Drawing straight line graphs

Finding the gradient of a straight line

Equation of a straight line: y = mx + c

Coordinates and ratio

Parallel and perpendicular lines

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Exponential graphs and circles - Higher

Trigonometric graphs - Higher

Solving equations using graphs

Graph transformations - Higher

Real-life graphs

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Velocity-time graphs - Higher

Gradients of real-life graphs - Higher

Algebra

Simplifying algebraic expressions

Multiplying and dividing algebraic expressions

Single brackets: Expanding and factorising

Double brackets: Expanding and factorising

Double and triple brackets - Higher

Solving equations

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Writing formulae and equations from word problems

Writing formulae and equations from diagrams

Rearranging formulae

Factorising quadratics

The quadratic formula - Higher

Complete the square - Higher

Algebraic fractions - Higher

Sequences

Finding the nth term

Solving inequalities

Inequalities on graphs - Higher

Iteration - Higher

Simultaneous equations: elimination and substitution

Non-linear simultaneous equations - Higher

Algebraic proof - Higher

Composite and inverse functions - Higher

Number

Types of numbers

Order of operations: BODMAS

Multiplying and dividing by powers of 10

Multiplying and dividing whole numbers

Multiplying and dividing decimals

Negative numbers: add, subtract, multiply, divide

Prime numbers and prime factorisation

Multiples, factors and prime factors

LCM and HCF

Fractions

Fractions, decimals and percentages

Writing recurring decimals as fractions

Rounding: Integers, decimal places, significant figures

Estimation

Error intervals

Upper and lower bounds - Higher

Powers and roots: Square and cube numbers

Laws of indices: multiply, divide, brackets

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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Learn with Basics

Learn the basics with theory units and practise what you learned with exercise sets!
Length:
Listing outcomes

Unit 1

Listing outcomes

Venn diagrams

Unit 2

Venn diagrams

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Optional

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Sets and Venn diagrams

Unit 3

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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