# Set Language and Notation

In these Secondary Math revision notes, you will learn set language and notation so that you will be proficient in solving Set Language exam questions.

You will learn how to:

- Understand Set Language
- Express words in Set Notation Symbols
- Understand relationships of different sets

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## Sets

A set is a collection of objects, things or symbols which are clearly identified.

The individual objects in the set are called the elements or members of the set.

Elements may be specified in two ways:

- by listing the elements
- by description

Example

Listing {2, 4, 6, 8, 10, 12} {A, E, I, O, U} | Description The set of even numbers between 1 and 13 The set of vowels in the alphabet |

{ } These braces stand for the word “the set of”

## Set Language and Notation

Set Language ‘… is an element of …’ ‘… is not an element of …’ The number of elements in set A Universal set The empty set Subset A is a (proper) subset of B A is not a (proper) subset of B Union of sets A and B Intersection of sets A and B Complement of set A Disjoint Sets | Notation ∈ ∉ n(A) U Ø ⊆ A ⊂ B A ⊄ B A ∪ B A ∩ B A‘ A ∩ B =Ø |

## Naming sets and number of members in a set:

Usually, we use Capital letters to denote a set and small letters to denote members of the set.

n ( ) indicates the total number of members in a set |

Example

A = {1, 3, 5, 7, 9, 11, 13} n (A) = 7

B = {2, 4, 6, 8, 10, 12} n (B) = 6

## Membership of the set:

∈ is an element of (is a member of ) means (belongs to) ∉ not an element of (is not a member of) means (does not belongs to) |

## Finite Sets

Sets in which all the elements can be listed.

A = {1, 3, 5, 7, 9} n (A) = 5

B = {days of the week beginning with S} n (B) = 2

## Infinites Sets

Sets in which it is impossible to list all the members of a set.

C = {2, 4, 6, 8, 10, . . . }

D ={x : x is a natural number}

## Set Language and Notation: Relation of Sets

**Universal sets:** U

The set which contains all the available elements.

All proper subsets formed within the universal set draw their elements from the available elements of the universal sets.

**Empty Set:** { } or ø

A set which contains **no** elements.

An empty set in a subset of any set.

**Equal Sets:** ⊆

If two sets have exactly the same elements, then we say that the two sets are equal sets.

Two equal sets are also subsets (denoted by ⊆) of each other.

Example If A = {2, 4, 6, 8} and B = {8, 6, 2, 4}

then A and B are equal sets, ie. A ⊆ B or B ⊆ A

**Subsets:**

A ⊆ B : A is a subset of B

When every element of set A is also an element of set B, then A is a subset of B.

: A is not a subset of C

There is at least one element in the first set that does not belong to the second set.

Example If A = {2, 4, 6, 8} , B = {8, 6, 2, 4} and C = {a, b, c, d}

then A ⊆ B

**Proper Subset:**

A B: A is a subset of B

When each element of set A is also an element of set B, but set B has more elements than set A, then set A is a proper subset of B, denoted by “A ⊂ B”.

Therefore, in this case set B is not a proper subset of A, denoted by “ B ⊄ A”

Example Given A = {1, 5, 9} and B = {1, 3, 5, 9}

then A ⊂ B , B ⊄ A

**Intersection of Sets:** ∩

Common elements in different sets.

A = {1, 2, 3, 4, 5, 6}

B = {1, 3, 5, 7, 9}

A ∩ B = {1, 3, 5}

**Union of Sets: **∪

The Union of set A and set B is the set of all elements which are in A, or in B, or in both A and B. It is denoted by ‘A ∪ B’ and is read as “the union of A and B”.

A = {1, 3, 4}

B = {5, 6, 7, 8}

A ∪ B = {1, 3, 4, 5, 6, 7, 8}

**Complement of a set: **A’

If ξ = {2, 3, 5, 7, 11, 13} and A = {2, 3, 7, 13}

A’ = {5, 11}

**Disjoint Sets:**

If the two sets have no element in common then the two sets are called disjoint.

The intersection of two disjoint sets is null or empty.

A = {1, 3, 5, 7}

B = {2, 4, 6, 8, 9}

A ∩ B = ø thus A and B are disjoint sets.

## Last Minute Revision for O Level Math?

## Check out our exam guide on other topics here!

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