Set Notation & Venn Diagrams (Cambridge (CIE) IGCSE Maths)

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Set Notation

What is a set?

  • A set is a collection of elements

    • Elements could be anything

      • Numbers, letters, coordinates, ...

  • You could describe a set by writing its elements inside curly brackets {}

    • {1, 2, 3, 6} , is the set of factors of 6

  • If the set of elements follow a rule then you can write this using a colon inside the curly brackets {... : ...}

    • The bit before the colon is the type of element

    • The bit after the colon is the rule 

      • {x is a positive integer : x2 < 30} is the set of positive integers which, when squared, are less than 30

      • This is equal to {1, 2, 3, 4, 5}

    • The colon is often read as 'such that' 

    • If no type is specified, x can take any value (fractions, decimals, irrationals, ...)

      • {x: x2 < 30} means any value whose square is less than 30

    • { (x, y) : y = mx + c } would mean the coordinates (x, y) where y = mx + c

      • I.e. The set of all possible coordinates that lie on the line y = mx + c

    • A colon can also be replaced by a vertical bar

      • {x | x2 < 30}

What do I need to know about set notation?

  • calligraphic E is the universal set (the set of everything)

    • For example, if we are only interested in factors of 24 then calligraphic E = {1, 2, 3, 4, 6, 8, 12, 24}

    • You may see alternative notations used for calligraphic E 

      • U is a common alternative (different to union for union!)

      • S or the Greek letter ξ (xi) may also be seen

  • We use upper case letters to represent sets (A, B, C, ...) and lower case letters to represent elements (a, b, c, ...)

  • n(A) is the number of elements in set A

    • For example, if calligraphic E = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, = {1, 4, 9}, = {1, 2, 3, 4, 5, 6}
      n(A) = 3, n(B) = 6

  • element of means "is an element of"

    • And not an element of means "is not an element of"

    • E.g. 3 element of open curly brackets 1 comma 2 comma 3 close curly brackets and 4 not an element of open curly brackets 1 comma 2 comma 3 close curly brackets

  • empty set is the empty set

    • This is the set which does not contain any elements

  • Asubset of or equal toB means "A is a subset of B"

    • Set A is a subset of set B if all the elements of A are also present in B

      • E.g. If A = {1,2,3} and B = {0, 1, 2, 3, 4}, then Asubset of or equal toB

    • A ⊈ B means "A is not a subset of B"

  • A intersection B  means the intersection of A and B (the overlap of A  and B)

    • This is the set of elements that are in both set A  and set B

      • E.g. If A = {1, 2, 3, 4, 5} and B = {-1, 1, 4, 7, 8}, then A intersection = {1, 4}

  • A union B  means the union of A  and B  (everything in A  or B  or both)

    • This is the set of elements that are in at least one of the sets

    • This includes elements in both sets (in the intersection)

      • E.g. If A = {5, 6, 7, 8} and B = {3, 7, 11}, then A union B  = {3, 5, 6, 7, 8, 11}

  • A' means the complement of A

    • It is the set of all elements in the universal set calligraphic E  that are not in A

      • E.g. If calligraphic E = {1, 2, 3, 4, 5} and A = {1, 3}, A'  = {2, 4, 5}

Sets & Venn Diagrams

What is a Venn diagram?

  • A Venn diagram is a way to illustrate all the elements within sets and any intersections 

  • A Venn diagram consists of

    • a rectangle representing the universal set (calligraphic E)

    • a circle for each set

      • Circles may or may not overlap depending on which elements are shared between sets

What do the different regions mean on a Venn diagram? 

  • A intersection B  is represented by the region where the A  and circles overlap

  • A union B  is represented by the regions that are in A  or or both

venn-diagram-2

Worked Example

Two sets and are shown in the Venn diagram.

Two sets, A and B, represented on a Venn diagram

(a) Write down n(A).

The elements of A are anything inside the A circle

A  = {2, 6, 12, 14, 28}

n(A) means the number of elements in A
There are 5 elements in A

n(A) = 5 

(b) Use set notation to complete the sentence {14, 28} = ...

 14 and 28 are the elements that are both in and B
This means they are in the intersection of and B

{14, 28} = B

(c) Write down the elements that are in set A U B.  

A = {2, 6, 12, 14, 28} and B = {7, 14, 21, 28, 35}
U is the set of elements that are in at least one of the sets 

For elements in both, only write them out once

A U = {2, 6, 7, 12, 14, 21, 28, 35}

(d) Jamie states that A'intersectionB' = empty set.

Explain if this statement is correct or not.

The statement means Not A and Not B is equal to an empty set
i.e. There are no elements that are Not A and Not B

However, there are 3 elements that are neither A nor B

There are 3 elements that aren't in A or B
A' intersection B' = {1, 5, 8}

Therefore the statement is incorrect

Worked Example

Consider the following sets.

calligraphic E = {letters of the alphabet}
T = {t, e, a, m}
I = { i }
M = {m, e}

Fill in the blanks with the appropriate set.

(i) ... space ⊈ space T

(ii) ... space subset of or equal to space...

The symbol subset of or equal to means "is a subset of", so all of the elements of one feature in the other
The symbol ⊈ means "is not a subset of"

The element in I is not in T

I space ⊈ space T

All the elements of M are also in T

M space subset of or equal to space T

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Mark Curtis

Author: Mark Curtis

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Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.

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Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.