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# Set Theory

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

otin S$.

**Example** − If 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

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