# Theory of Set; A Brief Note (Elementary Level)

## Introduction

The theory of sets was developed by a German mathematician Georg Cantor. The set theory plays a vital role in today’s many branches of mathematics, economics etc. In theory sets are used to define the concept of relations and functions. This hub briefly explains the set theory and related concepts.

__What is a Set ?__

By definition a set is one which contains many objects. These objects are called ‘elements of a set’.

For example

1) Even number 0 to 10. That is, 2,4,6,8

2) People of a country or a state

__How to Represent a Set?__

Suppose a set with the elements of natural numbers 1 to 10. The set can be represented with in braces { }. And further we can give a name for the set, suppose ‘N’. So the set can be represented as given below.

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

Here ‘N’ is the set and 1,2,3,4,5,6,7,8,9,10 are the elements which can be denoted by E.

__Some Concepts of Set__

Following are the some concepts related to the set theory.

__Two methods of Representing a Set__

Generally there are two methods to present a set.

1- Roster or tabular method

I the above example the set ‘N’ represented as tabular method. That is, N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

2- Set builder method

It is another method of representing a set. By this method the above set can be represented as given below.

N = {x/ x natural numbers between 1 to 10}

__Finite and Infinite Sets__

If the elements of a set are finite, it is a finite set.

For example – N= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Infinite set is one the elements are infinite.

For Example – N= {1, 2, 3, 4, 5, 6, 7,….. }

__Equal Sets__

Suppose consider two sets below,

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

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

Here the elements of set A are the same elements of set B. So, it will equal, it is called equal sets where set A = set B.

__Null Set__

Null set or empty set is a set, in which there are no any elements. It can be represented as { }. {0} is not a null set.

__Singleton Set__

If a set contain only a single element, it is called singleton set.

For example = { }, {0}, {17}

__Union Set and Sub Set__

Suppose consider set A= {a, b, c, d, e}, B = {a, d, e} and C= {b, c, e}.

Here A is the union set (u) and B and C are the sub sets. Because, few elements in set ‘A’ are included in the elements of set B and set C.

Union set and Subset can be represented in Venn diagram as showing below.

## Power Set

Power set of a set is the all possible subset of that set.

For example; A= {1, 2, 3}

Subsets (P) = {},{1},2},{3},{1,2},{1,3}{2,3},{1,2,3}

Here the number of elements of set A are 3 and its subsets are 8. Note that {} will be a subset of all sets. The number of subsets (power set) of a set can be calculated by raising the number of elements to 2.

For example, A= {a, b, c, d}

Since in the set A contain 4 elements total subsets (power set) will be 2*2*2*2 = 16 or 2^4 = 16.

__Two Major Set Operations__

Mainly, there are two operations can be seen in set theory. They are Union operation and intersection operation.

I) __Union Operation__

To find union of two sets, we have to join all the elements of two sets. But the elements never repeat.

For example;

A= {a, b, c, d, e, f}

B= {a, c, f, h, j, m}

Here Union of the set A and set B

Becomes AUB = {A, B, C, D, E, F, H, J, M}

II) __Intersection Operation__

To find intersection of two sets we have to take common elements in both sets. It is denoted by __inverse ‘U’__

For example;

A = {a, b, c, d, e, f} and B = {a, c, e, f}

Then A intersection B = {a, c, e, f}

__Compliment of a Set__

Compliment of a set contain elements outside the set or in universal set. It is denoted by A.

Suppose consider U= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

And A = {1, 3, 7, 9, 10}

Then A= {2, 4, 5, 6, 8}

__Differences of Two Sets__

It is the difference between two sets.

For example;

A = {1, 3, 5, 7, 9} and B = {2, 3, 6, 7, 9}

A - B = {1, 5}

B - A = {2, 6}

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