Set Notation & Venn Diagrams (Edexcel IGCSE Maths A (Modular): Higher Unit 1): Revision Note

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

What do I need to know about set notation?

• $\mathcal{E}$ 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}

• 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$\subset$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$\subset$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}

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 ($\mathcal{E}$)

  • a circle for each set

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

• A Venn diagram either shows:

  • the elements in each intersection

  • the total number of elements in each intersection

What do the different regions mean on a Venn diagram? 

• $A\cup B$  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

• The two circles intersect and split the rectangle into four regions

  • A region where both circles intersect

    • $A\cap B$

  • Two regions where each circle does not intersect with the other circle

    • $A\cap B\text{'}$ and $A\text{'}\cap B$

  • A region outside the circles

    • $A\text{'}\cap B\text{'}$

    • This can also be written as $\left(A\cup B\right)\text{'}$