# Set Theory

MathematicsComputer EngineeringMCA

German mathematician G. Cantor introduced the concept of sets. He had defined a set as a collection of definite and distinguishable objects selected by the means of certain rules or descriptions.

Set theory forms the basis of several other fields of study like counting theory, relations, graph theory, and finite state machines. In this chapter, we will cover the different aspects of Set Theory.

## Set - Definition

A set is an unordered collection of different elements. A set can be written explicitly by listing its elements using a set bracket. If the order of the elements is changed or any element of a set is repeated, it does not make any changes in the set.

### Some Example of Sets

• A set of all positive integers
• A set of all the planets in the solar system
• A set of all the states in India
• A set of all the lowercase letters of the alphabet

## Representation of a Set

Sets can be represented in two ways −

• Roster or Tabular Form
• Set Builder Notation

### Roster or Tabular Form

The set is represented by listing all the elements comprising it. The elements are enclosed within braces and separated by commas.

Example 1 − Set of vowels in English alphabet, A = { a,e,i,o,u }

Example 2 − Set of odd numbers less than 10, B = { 1,3,5,7,9 }

### Set Builder Notation

The set is defined by specifying a property that elements of the set have in common. The set is described as

A = { x : p(x) }

Example 1 − The set { a,e,i,o,u } is written as −

A = { x : x is a vowel in English alphabet }

Example 2 − The set { 1,3,5,7,9 } is written as −

B = { x : 1 ≤ x < 10 and (x % 2) ≠ 0 }

If an element x is a member of any set S, it is denoted by $x \in S$ and if an element y is not a member of set S, it is denoted by $y \notin S$.

ExampleIf S = {1, 1.2, 1.7, 2} , 1 ∈ S but 1.5 ∉ S

### Some Important Sets

N − the set of all-natural numbers = {1, 2, 3, 4, .....}

Z − the set of all integers = {....., -3, -2, -1, 0, 1, 2, 3, .....}

Z+ − the set of all positive integers

Q − the set of all rational numbers

R − the set of all real numbers

W − the set of all whole numbers