Set Notation & Venn Diagrams (Edexcel 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 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. {xx2 < 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. {(xy) : = 2x + 1} is the set of points that lie on the line y = 2x + 1

What do I need to know about set notation?

  • calligraphic E is the universal set (the set of everything)
    • e.g.  if talking about 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
      • S or the Greek letter ξ (xi) may also be seen
  • ∅ is the empty set (the set of with no elements}
    • e.g. {xx 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
  • aA 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}
  • AB means A is a proper subset of B
    • This means A is a subset of B but not the same as (A ⊆ but AB)
    • 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
    • AB means A is not a proper subset of B
  • AB 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
  • AB 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 (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 apostrophe is represented by the regions that are not in the A circle
  • A intersection B is represented by the region where the A and B circles overlap
  • A union B is represented by the regions that are in A or B or both

venn-diagram-2

Worked example

Two sets 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} = AB

 

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

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

A'∪= {1, 5, 7, 8, 14, 21, 28, 35}

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Dan

Author: Dan

Expertise: Maths

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.