Set Notation & Venn Diagrams | CIE IGCSE Maths: Extended Revision Notes 2023

Set Notation

What is a set?

A set is a collection of elements
- Elements could be anything - numbers, letters, coordinates etc
You could describe a set by writing its elements inside curly brackets {}
- e.g. {1, 2, 3, 6} is the set of the 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 just the type of element
- The bit after the colon is the rule
- e.g. {x : x² < 100} is the set of numbers which are less than 100 when squared
The elements are usually numbers but these could be coordinates
- e.g. {(x, y) : y = 2x + 1} is the set of points that lie on the line y = 2x + 1

What do I need to know about set notation?

$E$ is the universal set (the set of everything)
- e.g. if talking about factors of 24 then $E$ = {1, 2, 3, 4, 6, 8, 12, 24}
- You may see alternative notations used for $E$
  - U is a common alternative
  - S or the Greek letter ξ (xi) may also be seen
∅ is the empty set (the set of with no elements}
- e.g. {x : x is an even prime bigger than 5} = ∅ as there are no even primes bigger than 5
We normally 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
- e.g. n({1, 4, 9}) = 3
- Note n(∅) = 0 as there are no elements in the set but n({0}) = 1 as there is 1 element in the set
a ∈ A means a is an element of A (a is in the set A)
- e.g. If x ∈ {1, 4, 9} then x is either 1, 4 or 9
A ⊆ B means A is a subset of B
- This means every element in A is also in B
- e.g. {students in class Y that pass the exam} ⊆ {students in class Y}
A ⊂ B means A is a proper subset of B
- This means A is a subset of B but not the same as B (A ⊆ B but A ≠ B)
- The difference between ⊆ and ⊂ is similar to the difference between ≤ and <
- e.g. {1, 2, 3, 6} ⊂ {1, 2, 3, 4, 6, 8, 12, 24}
Putting a cross through the symbol means it is not true
- Similar to ≠ meaning not equal
- a ∉ A means a is not an element of the set A
- A ⊈ B means A is not a subset of B
- A ⊄ B means A is not a proper subset of B
A ∩ 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
A ∪ 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 sets
A' is “not A” (everything outside A)
- This is the set of elements that are not in A

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 ( $E$ )
- 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?

$A'$ is represented by the regions that are not in the A circle
$A \cap B$ is represented by the region where the A and B circles overlap
$A \cup B$ is represented by the regions that are in A or B or both

venn-diagram-2

Worked example

Two sets A and B are shown in the Venn diagram.

Venn-Set-1, downloadable IGCSE & GCSE Maths revision notes

(a)

Write down n(A).

The elements of A are anything that is inside the A circle. A = {2, 6, 12, 14, 28}. There are 5 elements in it.

n(A) = 5

(b)

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

14 and 28 are the elements that are in both A and B.

{14, 28} = A∩B

(c)

Write down the elements that are in set A'∪B.

A' is the set of elements not in A so A' = {1, 5, 7, 8, 21, 35}.
B = {7, 14, 21, 28, 35}.
A'∪B is the set of elements that are in at least one of the sets.

A'∪B = {1, 5, 7, 8, 14, 21, 28, 35}

Set Notation & Venn Diagrams (CIE IGCSE Maths: Extended)

Revision Note

Set Notation

What is a set?

What do I need to know about set notation?

Sets & Venn Diagrams

What is a Venn diagram?

What do the different regions mean on a Venn diagram?

Worked example

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Author: Dan