Set Notation & Venn Diagrams (Edexcel IGCSE Maths A): Revision Note

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

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}

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  = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A  = {1, 4, 9}, B  = {1, 2, 3, 4, 5, 6}
  n(A) = 3, n(B) = 6

means "is an element of"

• And means "is not an element of"

• E.g. and

is the empty set

• This is the set which does not contain any elements

AB 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 AB

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}

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

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

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