Set Notation & Venn Diagrams

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}

$\mathcal{E}$ is the universal set (the set of everything)

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

• You may see alternative notations used for $\mathcal{E}$ 

  • U is a common alternative (different to $\cup$ 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 $\mathcal{E}$ = {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

$\in$ means "is an element of"

• And $\notin$ means "is not an element of"

• E.g. $3\in \left\{1,2,3\right\}$ and $4\notin \left\{1,2,3\right\}$

$\varnothing$ is the empty set

• This is the set which does not contain any elements

A$\subseteq$B 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 A$\subseteq$B

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

A $\cap$ 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 $\cap$ B  = {1, 4}

A $\cup$ 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 $\cup$ B  = {3, 5, 6, 7, 8, 11}

A' means the complement of A

• It is the set of all elements in the universal set $\mathcal{E}$  that are not in A

  • E.g. If $\mathcal{E}$ = {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 ($\mathcal{E}$)

• a circle for each set

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

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

$A\cap B$  is represented by the region where the A  and B  circles overlap