# Sets

## Sets Revision

**Sets**

A** set** is a **collection** of numbers, usually defined by specific **rules** or **properties**. Each object in a set is called an **element**, or a **member**.

**Introduction**

We can use curly brackets to represent **sets**:

For example, we can represent the** set** of odd numbers, O, between 1 and 10 as…

O=\{1, 3, 5, 7, 9\}

We can also represent **sets** in **Venn diagrams**.

For example, let’s represent the **sets** A and B in this **Venn diagram**. A and B both have a circle containing their **elements**, and the intersection of them contains **elements** in both A and B.

Any **elements** outside of the circles do not lie in either set.

**Describing Sets**

There are some more ways of describing sets:

- The
**set**of numbers, x, such that x is less than 2

\{x:x<2\}

- N(A) means the number of
**elements**. For example, N(A)=12 means there are 12**elements**.

**The Universal Set and the Empty Set**

When using a **Venn diagram**, all** elements** in the diagram (including those in the circles and outside the circles), are called the** universal set**. From the **universal set**, we identify those that fit in A and B.

The **universal set** is represented by the letter, \xi, which is always present in the top left corner of **Venn diagrams**.

There is also the **empty set**, \emptyset, the set containing no elements. The empty set can also be written as the **complement** of **set** \xi.

## \boldsymbol{A} and \boldsymbol{A'}

The** set** A is represented by these sections of the **Venn diagram**:

This includes the **elements** only in A and those that are in in both **sets** A and B

Conversely, A' means **NOT** in A.

This means, all **elements** within the the **universal set** that is not in A, as represented by this **Venn diagram**:

**The Union of Two Sets**

The union of two** sets** means all** elements** either in A, B, or both. This can be written as…

A\cup B

**The Intersection of Two Sets**

The intersection of two sections contains **elements** that are in BOTH A and B. This can be written as…

A\cap B

**Subsets**

A subset is a** set** that is entirely contained within another **set**.

For example, if,

A=\{2,6,8\}, and,

B is the **set** of even numbers between 1 and 10, then,

A \subset B

We use the notation, \not\subset, if a set is not a subset of another set.

**Example 1: Sets**

A = \{1,4,6,8,10,11\}\\

B=\{2,4,8,10,11,12\}

Write down the** sets**

a) A\cup B

b) A\cap B

**[4 marks]**

a) A\cup B means the union of A and B, so all **elements** in either or both A and B.

So, A\cup B=\{1,2,4,6,8,10,11,12\}

b) A\cap B means the intersection between A and B, so all **elements** in both A and B.

So, A\cap B=\{4,8,10,11\}

**Example 2: Venn Diagrams**

Use the following** Venn diagram**:

to find,

a) C\cup D

b) C'

**[4 marks]**

a) C\cup D are the **elements** that are in either C or D. Therefore,

b) C' are the elements not in C. Therefore,

C'=\{2,3,5,6,1\}

**Example 3: Inequalities**

Solve the inequality 4x>16, expressing your answer in** set** notation.

**[3 marks]**

Let’s solve this inequality as usual,

4x>16

Divide through by 4 to isolate x,

x>4

We can write this in **set** notation,

\{x:x>4\}

## Sets Example Questions

**Question 1**: X and Y are sets, given by,

X=\{21,26,32,33,38,42,44\}\\

Y=\{11,25,32,34,40,44\}

Find the sets,

a) X\cup Y

b) X\cap Y

**[6 marks]**

a) The union of X and Y,

X\cup Y=\{11,21,25,26,32,33,34,38,40,42,44\}

b) The intersection of X and Y,

X\cap Y=\{32,44\}**Question 2**:

Find the sets,

a) C'\cup D'

b) C\cap D'

**[4 marks]**

a) The union of the elements not in C and not in D,

C'\cup D'=\{5,16,11,9,2,3,19\}

b) The intersection between the elements in C but not in D, which is simply the elements in C but not in D,

C\cap D'=\{11,9\}**Question 3**: Solve the inequality 12x-31<77, giving your solution as a set.

**[3 marks]**

Let’s solve the inequality as normal,

Add 31 to each side,

12x<108

Divide through by 12,

x<9

As a set,

\{x:x<9\}

**Question 4**:

X=\{1,3,5,7,9,11,13,15,17,19\}

Decide which set(s) of the following are subset(s) of X,

a) A=\{1,5,8,9\}

b) B=\{7,9,19\}

c) C=\{3,5,13,17\}

**[2 marks]**

A subset of X will contain only elements that also exist in X

a) Is not a subset as 8 is not in X

B and C are subsets of X as their elements are contained within X.

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