# 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.

## 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) is a subset of every 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 =  .

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).