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Sets and Venn diagrams

Sets and Venn diagrams

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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 11 and 1010:

  • 2,4,6,82,4,6,8 }
  • {x:x: xx 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(AA​)

Number of elements in set AA.


The empty set \emptyset​​​

A set which has nothing in it.


Universal Set ξ\xi or EE or UU​​

The set containing all elements.

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

Complement AA' or A\overline{A}​​

All elements not in AA.

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

Intersection ABA \cap B

All elements in both AA​ and BB.

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

Union ABA \cup B

All elements in either AA or BB or both.

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

Conditional (A given B) ABA|B​​

All elements in AA​ given they are in BB.

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 ABA \subset B

AA​ is a set inside of BB.

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)}= 100


Work out the number of students that like green.

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


Work out the probability that a student likes green.

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


2. Both blue and green:

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

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


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

P(BG)=20100=15\text{P(}B \cap G)= \frac{20}{100}=\frac{1}{5}


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

Note: In the example above 2525 is the number of students that did not fit into either category - they did not like blue or green!

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Exercises

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FAQs - Frequently Asked Questions

What are disjoint sets?

What is the union of two sets?

What is the intersect of two sets?

What does the complement of A mean?

What is the universal set?

How do I represent a set?

What is a set?

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