Set Notation & Venn Diagrams (Edexcel GCSE Maths)

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?

• is the universal set (the set of everything)

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

  • You may see alternative notations used for  

    • U is a common alternative (different to 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, ...)

• means "is an element of"

  • E.g.

• 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

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

• 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 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 B  = {3, 5, 6, 7, 8, 11}

• A' means the complement of A

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

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

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 ()

  • 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? 

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

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

How do I find probabilities from Venn diagrams?

• Count the number of elements you want and divide by the total number of elements

• For the Venn diagram shown below,

  • The probability of being in A  is 

    • There are 5 elements in A out of 11 in total

  • The probability of being in both A  and B  is

    • There are 2 elements in A  and B  (the intersection)

  • The probability of being in A, but not B, is 

    • 3 elements are in A  but not B

• Some harder questions are not out of the total number, but out of a restricted number

  • The probability of being in B, given that you are already in A, is 

    • You are only interested in elements in A

    • There are 5 elements in A, out of which only 2 are also in B