Set Theory, Sets, Notation, Representation, Types of Sets

# Set Theory, Sets, Notation, Representation, Types of Sets

## Set Theory

Set theory and its basic foundations were developed by Georg Cantor, a mathematician from Germany towards the end of the 19th century.

## Sets

A set is a collection of well-defined distinct objects.
The word ‘distinct’ means that the objects of a set must be all different.
The phrase ‘well-defined’ means that a set must have some specific property so that we can identify whether or not an object belongs to that set.
Now, let us observe the following collections:
1. All even natural numbers less than 13, i.e. 2, 4, 6, 8, 10, 12.
2. Prime factors of 30, i.e. 2, 3, 5.
3. All students of your class whose height exceeds 150 cm.
4. All months of a year.
The above collections are well-defined collection of objects, i.e. we can say whether an object belongs to the collection or not. So, the above collections form sets.
Now, consider the following collections:
1. Collection of brilliant students of your class.
It is not a well-defined collection as different people might have a different perspective on whether a student of your class is brilliant or not. For a student getting 40% marks, a student getting 60% is brilliant while a student getting 90% marks may call him an average student. So, it is not a set.
2. Collection of three months of a year.
It is not a well-defined collection as it is not known which three months of a year are to be included in the collection. So, it is not a set.

## Notation of a Set

We usually denote sets by capital letters and their elements by small letters.
All the elements of the set are enclosed in curly brackets { } and are separated by commas (,).
Consider the set of odd natural numbers less than 10, i.e. 1, 3, 5, 7, 9.
Let us call this set as A. Then, A = {1, 3, 5, 7, 9}
Thus, the given set is a well-defined collection of numbers. Also, no two numbers in it are identical.

## Representation of a Set

A set can be represented by the following methods.
i. Description method
ii. Roster method or tabular form
iii. Rule method or set builder form

### 1. Description Method

In this method, a well-defined description of the elements of the set is made. The description of elements is enclosed in curly brackets.
For example, the set of natural numbers less than 10 is written as A = {natural numbers less than 10}.

### 2. Roster Method or Tabular Form

In this method, we list out all the elements of the set in curly brackets and separate them by commas.
For example, the set of 2-digit numbers whose sum of the digits is 9 is written as B = {18, 27, 36, 45, 54, 63, 72, 81, 90}.

### 3. Rule Method or Set Builder Form

In this method, we write a variable representing any member of the set followed by a property, rule or a statement satisfied by each member of the set and enclose it in curly brackets.
For example, the set of factors of 36 is written as C = {x : x is a factor of 36}.

## Cardinal Number of a Set

The number of elements in a set is called the cardinal number of a set. It is denoted by n(A).
For example, if A = {2, 4, 6, 8, 10}, then n(A) = 5.

## Types of Sets

### Finite Set

If the elements of a set can be counted, the set is called a finite set.
For example, the set of natural satellites of Jupiter and the set of two-digit prime numbers.

### Infinite Set

If the elements of a set cannot be counted, the set is called an infinite set.
For example, the set of fractions lying between 1 and 2 and the set of integers less than 10.

### Empty Set

A set that contains no element is called the empty set. It is denoted by the symbol { } or ϕ.
For example, the set of prime numbers between 5 and 7.

### Singleton Set

The set which contains only one element is called a singleton set.
For example, A = {the number of stars in our Solar System} and B = {2}

### Equal Sets

Two sets A and B are said to be equal if they have same elements. It is written as A = B.
For example, if A = {1, 3, 5, 7} and B = {5, 7, 3, 1}, then A = B, because the elements of A and B are same.

### Equivalent Sets

Two sets A and B are said to be equivalent sets if they have the same number of elements, the elements may or may not be the same. Thus, two sets A and B are equivalent if n(A) = n(B).
For example, if A = {colours of the rainbow}, B = {x : x is a prime number less than 19}, then n(A) = 7 = n(B).

### Overlapping Sets

Two sets A and B are said to be overlapping sets if they have at least one element in common.
For example, if set A = {letters of the word DELHI} and B = {letters of the word BHOPAL} are overlapping because the letters L and H are common in both the sets.

### Disjoint Sets

Two sets A and B are said to be disjoint sets if they have no element in common.
For example, the set A = {x : x is a student of ABC school} and set B = {x : x is a student of XYZ school} are disjoint sets as no student can study in both the schools at any one point.

## Complement of a Set

The complement of a set A is the set of all elements in the universal set which are not in set A.  It is denoted by A’ and is read as ‘complement of A’.
Thus, if universal set is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 4, 6, 8, 10}, then A′ = {1, 3, 5, 7, 9}.

Solved Examples:

Example 1: Express each of the following sets in the form as required.

a. The set of integers between –3 and 3 (Roster method)

b. C = {x : x is a letter in the word LOLLIPOP} (Description method)

Solution:

a. {–2, –1, 0, 1, 2}

b. C = {letters of the word LOLLIPOP}

Example 2: Write the following sets in set builder form.

a. The set A of even natural numbers lying between 5 and 20.

b. B = {10, 20, 30, 40, …}

Solution:

a. A = {x | x = 2k, 3 ≤ k ≤ 9, k ϵ N}

b. B = {x | x = 10n, n ϵ N}

Example 3: Write the following sets in roster form.

P = {x : x = n2 + 2, n ϵ N and n ≤ 5}

Solution:

P = {3, 6, 11, 18, 27}

Example 4: Find the cardinal number of the following sets.

a. A = {x : x n2 + 1, n ϵ N and n ≤ 4}

b. B = {x : x is a day of a week}

c. D = {x : x is a letter of the word ‘NATIONAL’}

Solution:

a. Here, A = {2, 5, 10, 17} which has 4 elements.

Hence, n (A) = 4

b. Here, B = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}

Hence, n (B) = 7

c. Here, D = {N, A, T, I, O, L}

Hence, n (D) = 6

Example 5: State whether each of the following statements is true or false when

A = {letters of the word ‘NUMERAL’} and B = {letters of the word ‘MATERIAL’}.

a. A and B are equal sets.

b. A and B are equivalent sets.

c. A and B are disjoint sets.

Solution:

Here, A = {N, U, M, E, R, A, L} and B = {M, A, T, E, R, I, L}

a. False as the elements are not identical

b. True as n (A) = n (B) = 7

c. False as elements M, E, R, L and A are in both the sets.