# Set Language And Notation

**Set:**

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

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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

E.g.

**Listing Description**

{2, 4, 6, 8, 10} The set of even numbers between 1 and 11

{a, e, i, o,u} The set of vowels in the alphabet

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

e.g. {Even numbers between 1 and 11}

### 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.

E.g.

A= {1, 3, 5, 7, 9, 11} n (A) = 6

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

**Membership of the set:**

Є is an element of (is a member of ) (belongs to)

is not an element of (is not a member of) ( does not belongs to)

E.g.

A = {1, 3, 5, 7, 9, 11}

**Finite Sets:**

Set in which all the elements can be listed.

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

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

**Infinite Sets:**

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

C= {2, 4, 6, 8,10….}

E={x:x is a natural number}

**Relations Of Sets:**

**Universal Sets: (Ƹ)**

The set which contains all the elements.

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

**Complement Of a Set: (A****`)**

(Ƹ element – A element} = A**`**

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

A**` **= {5, 11}

**Equal Sets:(C)**

Two sets A and B are said to be equal if and only they have exactly the same elements.

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

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

Then

A=B B=A A __C__ B B __C__ A

**Subsets:**

When each member of a set A is also a member of a set B, then A is a subset of B.

C** is a subset** **of :**When two sets have exactly same elements or elements in the first set are also elements in the second set.

**is not a subset** **of:** There is at least one element in the first set that does not belong to the second set.

**Proper Subset:**

When each member of a set A is also a member of a set B, but set B has MORE elements than set A, then set A is a proper subset of B, denoted by “A C B”. Therefore, in this case set B is not a proper subset of A (B A)

**C is a proper subset of:** When each element in the first set also belongs to the second set, but the second set has more elements than the first set.

**C is not a proper subset**: When there is at least, one element in the first set that does not belong to the second set.

E.g. A = {1, 5, 9} B = {1, 3, 5, 9}

**Empty Set: ({} or )**

A set which contains **NO** elements

An empty set in a subset of any set.

**Intersection Of Sets: ()**

Common elements in different sets.

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

B= {2,4,8,10}

**Union Of Sets: U**

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 U B’ and is read as “the union of A and B”.

A= {1,3,4} B={6,7,8}

A U B = {1,3,4,6,7,8}

**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.

e.g. A = {1,3,5,7} and B = {2,4,6,8,9}

A**B = thus A and B are disjoint sets.**

**De Morgans Law:**

**Venn Diagram:**

In a venn diagram, we use a rectangle to denote a universal set Ƹ and a loop such as circle or an oval to represent any set in Ƹ .

**Examples of Venn Diagrams:**

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