The Theory of Sets Basic Concepts

## Introduction to the Theory of Sets

- German mathematician
**Georg Cantor**developed the theory of sets in the end of the 19th century. - Sets are used to define the concepts of
**relations**and**functions**. - Sets are used in the study of
**geometry, sequences, probability**, etc.

## Definition of Sets

- A collection is a group of objects of a particular kind.
- A
**well-defined collection**of objects is called a**Set**. - Any element (a) belongs to set A is represented by a ∈ A.
- n(A) represents the number of elements in a set A.

Examples of Collections:

- A group of cricket players.
- A group of few playing cards.
- A group of good students.
- A group of some positive numbers.

Examples of Sets:

- All the vowels in the English Alphabet.
- All the prime numbers below 10.
- All the cricket players with centuries more than 20 in the ODIs.
- All the blacks colored cards in a pack of playing cards.
- All the natural numbers (N).
- All the rivers of India.
- All kinds of triangles.

## Representation of Sets

**1. Roster form / Tabular form / List form**

- All the elements of a set are shown as a list enclosed in { }.
- The elements are separated by commas.
- The order of the elements does not matter.
- Repetition of the elements is not allowed.
- For example,
**A set of even numbers less than 10 = {2, 4, 6, 8}**

** **

**2. Set-builder form**

- In the set-builder form, sets are represented by a unique common property of all the elements.
- Set is defined by the definition of the unique common property.
- This definition is enclosed by { }.
- For example,
**S = {x : x is an even natural number and x < 10}**

#### Roster form ⇔ Set-builder form

{ ½, ⅓, ¼, ⅕ } = {x : x = 1/(n+1); where n ∈ N and n ≤ 5}

### How to read the Set-builder form of a set?

Let us take a set A as shown:

A = {x : x = 1/(n+1); where n ∈ N and n ≤ 5}Then,

A is a set of all x such that x = 1/(n+1) where n is a natural number equal or less than 5.

A = {x : x = 1/(n+1); where n ∈ N and n ≤ 5}Then,

A is a set of all x such that x = 1/(n+1) where n is a natural number equal or less than 5.

## Types of Set

### Empty Set

- It does not contain any element in the set.
- It is also called a
**void set**. - The empty set is denoted by the symbol
**∅**or**{ }.** - n(A) = 0.

** **

**Non-empty Set**

- It contains at least one element in the set.
- The number of elements can be finite or infinite.
- n(A) ≠ 0.

**Finite Set**

- It contains a definite or finite number of elements.
- The number of elements can be counted.
- An empty set is also an example of an empty set.
- n(A) = finite number.

### Infinite Set

- The number of elements can not be counted.
- The number of elements is not definite/finite.
- n(A) ≠ finite number.
- Infinite sets cannot be described in the roster form.

### Equal Sets

- Two sets A and B are said to be equal if they have exactly the same elements.
- Equal sets are written as
**A = B**.

### Unequal Sets

- Two sets A and B are called unequal sets if they are not equal sets.
- Equal sets are written as
**A ≠ B**.

## Subset

- Set (
**A)**is said to be a subset of a set**(B)**if every element of set (A) is also present in the set (B). **A ⊂ B if a ∈ A ⇒ a ∈ B**.- ‘⇒’ means ‘that implies’.
- A ⊂ B ⇒ A is a subset of B.
- A ⊄ B ⇒ A is not a subset of B.
**∅**(an empty set)- The set (A) itself is also a subset of A. (
**A ⊂ A)**

## Universal Set (U)

- A universal set is a set of all the elements or objects in any context.
- Every set in that context is a subset of U.

## Power Set – (Set of all the subsets of set A)

- The collection of all subsets of a set A is called the power set of A.
- It is denoted by P(A).
- In P(A), every element is a set.

### How to find the number of elements in a Power Set?

>> For example, A = {1, 2}

>> n(A) = 2

>> Subset of A = { }, {1}, {2}. {1,2}

>> P(A) =

**{**{ }, {1}, {2}. {1,2}**}**>> Number of elements in the Power set of A = n(

**P(A)**) = 4 = 2².

In general, if n(A) = m then n(

**P(A)**) = 2m.

## Some Important Sets of Numbers

**Natural Numbers (N)**= A set of all-natural numbers.**Whole Numbers (W)**= A set of zero and all natural numbers.**Integers (I) or (Z) =**All the whole numbers along with their negatives.**Rational Numbers (Q) =**All the numbers that can be expressed as a fraction or a quotient (p/q form) is called a Rational Number where (q ≠ 0) and (p & q) both are co-prime numbers i.e. (p & q) don’t have any common factor.**Irrational Numbers (T)**= All the real numbers that are not rational.**Real Numbers (R)**= All the rational and irrational numbers.

### Examples of subsets

- N ⊂ W ⇒ N is a subset of W
- W ⊂ I ⇒ W is a subset of I
- I ⊂ Q ⇒ I is a subset of Q
- T ⊂ R ⇒ T is a subset of R
- Q ⊂ R ⇒ Q is a subset of R

In short, N ⊂ W ⊂ I ⊂ Q ⊂ R.

## Intervals as subsets of R

a and b are two real numbers where a < b. Below are the types of intervals.

- Closed interval = [a, b]
- Open interval = (a, b)
- Semi Open interval = (a, b] and [a, b).

### How to write intervals in the form of sets?

- [a, b] is a set of all the real numbers between a and b. (including a and b)
- (a, b) is a set of all the real numbers between a and b. (excluding a and b)
- [a, b) is a set of all the real numbers between a and b. (including a but excluding b)
- (a, b] is a set of all the real numbers between a and b. (excluding a but including b)

Clearly, all these sets are subsets of Set R (the set of all the real numbers).