Set Notation & Venn Diagrams (Cambridge International IGCSE Mathematics)

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?

• $E$ is the universal set (the set of everything)

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

  • You may see alternative notations used for $E$

    • 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 $E$ = {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. $3 \in \{ 1 , 2 , 3 \}$ and $4 \notin \{ 1 , 2 , 3 \}$

• $\emptyset$ 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 $E$ that are not in A

    • E.g. If $E$ = {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 ($E$)

  • 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?

• $A\cup B$ 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

Worked Example Two sets A and B are shown in the Venn diagram.

(a) Write down n(A).

Answer:

The elements of A are anything inside the A circle

A = {2, 6, 12, 14, 28}*

n(A) means the number of elements in A There are 5 elements in A

n(A) = 5 **

(b) Use set notation to complete the sentence {14, 28} = ...

Answer:

• 14 and 28 are the elements that are both in** A and B* *This means they are in the intersection of A and *B

**Final answer:****{14, 28} = **A **∩ B

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

Answer:

A* = {2, 6, 12, 14, 28} and B = {7, 14, 21, 28, 35}* A* U B is the set of elements that are in at least one of the sets *

For elements in both, only write them out once

Final answer: A U B = {2, 6, 7, 12, 14, 21, 28, 35}

(d) Jamie states that equationA'$\cap$B' = $\emptyset$.

Explain if this statement is correct or not.

Answer:

The statement means Not A and Not B is equal to an empty set *i.e. There are no elements that are Not A and Not *B

However, there are 3 elements that are neither A nor* B*

There are 3 elements that aren't in A or B equationA' $\cap$ B' = {1, 5, 8}

Final answer: Therefore the statement is incorrect

Worked Example Consider the following sets.

equation$E$ = {letters of the alphabet} $T$ = {t, e, a, m} $I$ = { i } $M$ = {m, e}

Fill in the blanks with the appropriate set.

equation(i) ... ⊈ T

(ii) $...\subseteq...$

Answer:

The symbol $\subseteq$ means "is a subset of", so all of the elements of one feature in the other The symbol ⊈ means "is not a subset of"

The element in $I$ is not in $T$

equation``I ⊈ T

All the elements of $M$ are also in $T$

$M\subseteq T$