# Operations performed on the set in Discrete Mathematics

In this article, we will learn about the cardinality of set, some standard notation of set, construction of set and the operations performed on the set in discrete mathematics.
Submitted by Prerana Jain, on August 12, 2018

Prerequisite: Set theory and types of set in Discrete Mathematics

### Cardinality of set

It is the number of elements in a set denoted like, A= {1, 2, 3, 4}

```    |A| = 4  | | symbol of cardinality
```

### Standard notations of set

There are many notations which are used in the set.

• C = set of all complex number that can be represented in form of a+ib, a, b are real number.
• R = set of all real number that is represented along a line.
• Q = set of all rational number that can be expression.
• Z = set of all integer.
• Number without fractional components.
• W = set of all whole number. Set of all positive (integer) and zero.
• N = Non negative number.

### Construction of set

A set can be represented by two methods:

1. Tabulation/Roster method
In the tabulation method, we describe a set by actually writing every number of the set. In this method, we prepare a list of objects forming the set writing the elements one after another between a pair of curly brackets. Thus a set a whose elements are 1, 3, 5,... will be written as A = {1, 3, 5, ...}.
2. Set builder
In this, we describe a set by actually writing the member of set the properties based on which reader can understand what is the set.
A = { x: x E Z 0<x<5 }

### Operations of set

There are many operations which are performed on the set:

1. Union of set

For any two set A and B, the union of A and B written as AUB is the set of all elements which are members of the set A or the set B or both, Symbolically it is written as: A U B = { x:x E A or X E B }

Example:

```
Let, A= {1, 2, 3, 4}
B = { 2, 4, 6, 8, 10}
Then , A U B = {1, 2, 3, 4, 5, 6, 7, 8, 10}

```

2. Intersection of set

The intersection of two set A and B denoted A intersection B is the set of elements which belongs to both A and B, it is written as: A intersection B = { x:x E A and xEB}

3. Difference of two set

The difference of two sets A and B in that order is the set of elements which belongs to A, but which O does not belong to B. We denote the difference of A and B by: A – B or A ~ B

Which reads as "A difference B" or "A minus B" symbolically A - B = {x:x E A and X does not belong to B}

A ~ B is also called the compliment of B with respect to A.

Example:

```
let,A = {a, b, c, d, e}
B= {f, b, d, g}
A – B = {a, c, e}
B – a = {f.g}
A – B does not equal to B – A.

```

4. Complement of set

If U is a universal set containing A then U-A is called the complement of A and is denoted by A' or Ac thus
A' = U – A = {x:x E U and x does not belongs to U}
A' = { x:x does not belong to A}

Example:

```
let,    U  = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A  = {2, 4, 6, 8, 9}
then    A' = {1, 3, 5, 7}

```

5. Symmetric difference

If A and B are two sets we define their symmetric difference as the set of all elements that belong to A or to B but not to both A and B and we denote it by (A - B)U(B – A).