# Set Notation

## Set Notation Revision

**Set Notation**

Understanding set notation is important when it comes to working with probability. Sets are usually denoted by a singular capital letter, and are either written down as a list of all numbers within that set or by giving a rule.

**Definition of Sets**

A **set** can be described by listing all the elements within it e.g. \{1, 3, 5, 7, 9\}. However, for sets with a large amount of elements this is often impractical, so these sets would often be defined by giving a **specific rule** e.g. \{\text{multiples of }3\}.

The standard notation to show that you are defining a set is to use **curly brackets** \{\}.

**Capital letters** are often used to denote the whole set, e.g. K=\{\text{all positive multiples of }2\}. Then a **lower case letter** would be used to denote an element within this set, e.g. x is an element of K.

Therefore, in this case this set could be described as K=\{x:x \text{ is a positive multiple of } 2\}.

The **universal set**, denoted by the Greek letter ‘**xi**‘ or \xi, which is the set that includes all the elements that the elements of another set can be selected from.

**Example: **

\xi=\{1,2,3,4,5,6,7,8,9,10\}

P=\{\text{prime numbers}\}

Then the elements of P are all the numbers within \xi that are **prime**, which are 2, 3, 5 and 7.

**The Union and Intersection of Sets**

The **union** of two sets can be defined as a set containing **all the elements in both sets**. This is denoted by a ‘**u**‘ shaped symbol, \cup, so you would write the **union** of set A and set B as A\cup B.

The **intersection** of two sets only **includes the elements that are in both sets**. This is denoted by a ‘**n**‘ shaped symbol, \cap, so you would write the **intersection** of set A and set B as A\cap B.

**Example: **

X=\{5, 10, 15, 20, 25\}

Y=\{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\}

X\cup Y=\{2, 4, 5, 6, 8, 10, 12, 14, 15, 16, 18, 20, 25\}

X\cap Y=\{10, 20\}

10 and 20 **appear in both sets**, so these are the only elements in the **intersection **of the two sets.

For the **union**, elements that **appear in both sets should only be listed once**.

**Defining Sets Using Algebra and Inequalities **

Sets will often be defined using **algebra **and/or **inequalities**, so you will need to decipher what the **description **of the set is before being able to work with it.

For example, a set A may be written as \{a:a>2\}, which means that the set A is the set of numbers, a, such that a is **greater** than 2.

Another example of a description of a set is \{(x,y):y=3x-1\}, which is the set of all the points (x,y) such that y=3x-1.

**Example:**

\xi=\{\text{positive integers}\}

N=\{x:x<5\}

List all the elements of the set N.

The elements of N **must also be contained within the universal set**, so the numbers which satisfy **both** definitions are:

1,2,3 and 4

**More Set Notation**

To express whether **something is an element of a set or not**, the symbols \in or \notin are used.

\in = something is an element of a set – e.g. 3 \in \{\text{odd numbers}\}

\notin = something isn’t an element of a set – e.g. 6 \notin \{\text{multiples of }5\}

If a set doesn’t have any elements contained within it, then it is called the **empty set**, denoted as \{\} or \varnothing.

For example, if set Z=\{\text{prime numbers that are multiples of }4\}, then set Z=\varnothing.

To express how many elements there are within a set, the notation n(A) is used, e.g. n(A)=8 shows that there are 8 elements within the set A.

**Using Venn Diagrams to Represent Sets**

Venn diagrams can be really helpful to visualise sets, especially the union and intersection of them.

Each set is represented by a circle, with all circles contained within a larger rectangle, which represents the universal set \xi.

The circles in a Venn diagram are labelled by a letter, which also corresponds to the set. For sets representing the quantity of something, e.g. the number of girls in a class, a number will be inside the circle.

**Intersection and Union**

The intersection of sets is represented as where the circles overlap, if the two sets don’t have any shared elements then the circles will not overlap.

The union of sets is represented by the entire space of the circles.

**Example:**

\xi=\{\text{Positive integers less than }20\}

C=\{\text{Multiples of }3\}

D=\{\text{Multiples of }2\}

Using the Venn diagram, we can see that

C \cup D=\{2,3,4,6,8,9,10,12,14,15,16,18\}

C \cap D=\{6,12,18\}

All elements that are within the universal set, \xi, but are not contained within either set C or set D are written outside of the circles.

**Complement and Subsets**

The **complement** of a set is all the elements of the **universal set** which **aren’t contained within the set**. The complement of a set G is written as G'.

**Example:**

\xi=\{10,11,12,13,14,15,16,17,18,19 \}

G=\{\text{even numbers}\}

The elements of G' are numbers within \xi that are not even, which are 11,13,15,17 and 19

**Subsets are sets that are completely contained within another set**, so all the elements of the subset are elements of the other set.

A ‘**proper subset**‘ is defined as a subset which has **less elements** than the set it is contained within.

The following notation is used for subsets:

A \subseteq B = A is a subset of B

A \nsubseteq B= A isn’t a subset of B

A \subset B= A is a proper subset of B

A \not\subset B= A isn’t a proper subset of B

**Example:**

\xi=\{\text{positive integers}\}

E=\{\text{positive even numbers}\}

Then we can write E\subseteq \xi

**Venn Diagrams – Complements and Subsets**

The **compliment** of a set is represented by everything **outside** of its corresponding circle on a Venn diagram. For example, \textcolor{#bd0000}{A'} is represented by everything outside of the circle for the set \textcolor{#bd0000}{A}.

**Proper subsets** are represented on Venn diagrams by a smaller circle which is **completely contained** within a larger circle. For example, the diagram on the right shows that \textcolor{#00bfa8}{B} is a proper subset of \textcolor{#00bfa8}{A}.

**Example 1: Union and Intersection**

Let A=\{3,6,9,12,15,18,21,24,27\} and B=\{2,4,6,8,10,12,14,16,18,20,22,24\}

Write down the elements in

**a)** A\cup B

**b)** A \cap B

**[2 marks]**

**a)** A \cup B contains elements from **at least one of** set A **or** set B.

Therefore, we can write A\cup B=\{2,3,4,6,8,9,10,12,14,15,16,18,20,21,22,24,27\}

Even though 6, 12, 18 and 24 **appear in both sets**, we only included them **once **within the set A\cup B

**b)** A \cap B contains elements that appear in **both** set A **and** set B.

Therefore, we can write A\cap B = \{6,12,18,24\}

**Example 2: Algebra and Inequality Notation**

**a)** Let \xi=\{\text{positive integers}\} and X=\{x:x<8\}

Write down all the elements within X.

**b) **Let \xi=\{\text{all 2D coordinates}\} and Y=\{(x,y):y=2x-3\}

Decide whether the following statement is true or false.

(6,8) \in Y**[4 marks]**

**a)** We know the **universal set** is all positive integers, therefore the elements in set X have to be **greater** than 0.

Thus, the elements within set X must be 1,2,3,4,5,6 and 7

**b)** For (6,8) to be an element of Y, then x=6 and y=8 would need to satisfy the equation y=2x-3

2(6)-3=9 \neq 8 so therefore (6,8) \in Y is **false**.

**Example 3: Complement and Subsets**

Let \xi=\{\text{positive integers less than }20\}, M=\{x:4\leq x \leq 9\}, N=\{\text{Factors of }24\}

**a)** Find M\cap N'

**b)** Find (M \cap N)'

**c)** Let O=\{\text{even numbers}\}

Is (M\cap N) \sub O a **true** statement?

**[6 marks]**

**a) **M=\{4,5,6,7,8,9\} and N=\{1,2,3,4,6,8,12\}

So M\cap N'=\{4,5,6,7,8,9\}\cap \{5,7,9,10,11,13,14,15,16,17,18,19\}=\{5,7,9\}

**b)** M \cap N= \{4,5,6,7,8,9\} \cap \{1,2,3,4,6,8,12\}=\{4,6,8\}

So (M \cap N)'=\{1,2,3,5,7,9,10,11,12,13,14,15,16,17,18,19\}

**c) **From b) we have that M \cap N=\{4,6,8\}

Therefore as all the elements of this set are **even**, (M\cap N) \sub O is true.

## Set Notation Example Questions

**Question 1:** Let \xi=\{\text{negative integers greater than }-20\} and T=\{\text{multiples of }3\}

Write down all the elements that are in the set T.

**[2 marks]**

The set T can only have elements that are the universal set, so it contains all multiples of 3 greater than -20 and less than 0.

So set T=\{-3,-6,-9,-12,-15,-18\}

**Question 2**: Let F=\{\text{positive even numbers less than }20\} and G=\{\text{positive multiples of }3\text{ less than }20

Write down the elements in

**a)** F \cup G

**b)** F \cap G

**[2 marks]**

**a)** The set F\cup G contains all the elements in both F and G.

So F \cup G=\{2,3,4,6,8,9,10,12,14,15,16,18\}

**b)** The set F\cap G contains the elements that appear in the set F and the set G.

So F\cap G=\{6,12,18\}

**Question 3:** Let \xi=\{\text{negative integers}\} and N=\{n:n\geq -5\}

Write down all the elements in the set N.

**[2 marks]**

We know that the universal set is all the negative integers, so all the elements within the set N must be less than 0.

Therefore, set N=\{-5,-4,-3,-2,-1\}

**Question 4:** Let \xi=\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15\}, P=\{x:7<x\leq 12\}, Q=\{\text{Multiples of }4\}

**a)** Find P'\cap Q

**b)** Let R=\{\text{Factors of }990\}

Show that (P\cap Q')\subseteq R

**[5 marks]**

**a)** We know that P=\{8,9,10,11,12\} and Q=\{4,8,12\}

So P'\cap Q=\{1,2,3,4,5,6,7,13,14,15\}\cap \{4,8,12\}=\{4\}

**b)** R=\{1,2,3,5,6,9,10,11,15\}

P=\{8,9,10,11,12\}

Q'=\{1,2,3,5,6,7,9,10,11,13,14,15\}

So P\cap Q'=\{9,10,11\}, which is a subset of R.

Also, n(P\cap Q')<n(R), which must be true because (P\cap Q')\subseteq R