# What are Sets? Types, Properties and solved questions

In this post we will discuss the basic definition and types of sets with example. The concepts has been explained in simple manner so that student with even weak Math background can grasp the concept, but still if you face any difficulty, feel free to contact regarding the same.

## Concept of Sets

### What is Set?

Set basically represents collection of objects.

In practical life you used different things for object collection.
For example:
–> you use wallet to keep the collection of money
–> we use plastic box to keep the pack of cards
This collection of objects with similar properties can be termed as sets.

Similarly, in mathematics, we use sets to collect objects with similar properties.

Consider the following set of even number below 10
N = {2,4,6,8,10}

==> N is a set name
==> { 2, 4, 6, 8, 10 } are elements of set N

Hence the above expression is a set N, which has kept the object 2, 4, 6, 8 and 10 inside.

### Set Representation and Properties

Here we will understand the right way to represent any set and also its important properties

#### 1. Set Representation

The set is always denoted by capital letter and the object is always in small letters

Set of even number below 10
N = {2,4,6,8,10}
The set is denoted by capital letter N

Set of vowels in English Alphabet
V = {a, e, i, o, u}
This is set V, whose objects are a, e, i, o, u

#### 2. Use of Epsilon \varepsilon

Set of even numbers below 10
N = {2, 4, 6, 8,10}
Here N is the name of the set and 2, 4, 6, 8,10 are the objects/elements

We can express individual object with the use of epsilon \varepsilon
For example, from above we can see that 2 is one of the element of N, so this can be expressed as:
2\varepsilon \ N , means “2 belongs to N

Set of vowels in English Alphabet
V = {a, e, i, o, u}
Hence a, e, i, o, u are the elements of set V

so we can say that i \varepsilon \ V
That is “i” belongs to set V

#### 3. Representation of Set

Set can be represented in two forms
a. Roster Form
b. Set Builder Form

a. Roster Form

In Roster, all the elements of the set are mentioned and they are separated with the help of commas.

Set of even number below 10 using Roster Form
E = { 2, 4, 6, 8}
There are four even numbers below 10 (2, 4, 6, 8), all have been mentioned separately in the set E

Set of numbers below 30 which are divisible by 7
N = {7, 14, 21, 28}

All the numbers are distinctively written in the set, this is the characteristics of Roster Method

Note 01:
There is no compulsion of order of elements
N = {7, 14, 21} can also be written as N = { 14, 7 , 21};
V = { a, e, i, o, u } can also written as

Note 02:
Elements are not repeated in Roster form
For example, if you are asked to write set of “APPLE”, it will be written as
N = { A, P, L, E}
Note that P is written only once

b. Set Builder Form

This is another way with the help of which set can be represented.
Here you don’t need to write all the element, just express them with the help of common property

Example 01
Set of vowels in English Alphabet
a, e, i, o, u are the vowels of English alphabets.
This can be represented in the form of set builder form as follows:

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

Hence we use the symbol x to represent all the element
and then we define the property of the element present

Example 02
Set of number below 30 which are divisible by number 7

Numbers that are divisible by 7 are (7, 14, 21, 28)
we can write the set as:
A = { x : x is divisible by 7 and x<30)

I hope you have now understood what sets are and how they are represented in technical ways.

### Different types of Sets

#### 1.Empty Set

Any set which do not contain any element is known as empty set/null set/void set

Example 01
Set of people living on Mars
Y = {x : x is number of people living in Mars}

Here Y is a set representing number of people living in Mars.
we know that today no human lives in mars, so the human population of mars = 0

So there is no element in set Y, hence it is empty set

Example 02
Set of natural number between 1 & 2
A = { x : x is a natural number and 1< x <2}

This set is also an empty set because there is no natural number which lies between 1 and 2.

Example 03
Set of people with more than 150 years of age
A = { x : x is the number of people of age greater than 150}

This set is also an empty set as there is no human in earth whose age is more than 150

#### 2. Finite sets

Any set which contains definite/fixed number of element is Finite set

Example 01
X = { 2, 3, 8, 9, 12}
The set X has fixed set of elements, hence it is a finite set

Example 2
Number below 30 which is divisible by 7
Y = {x : x is divisible by 7 and x< 30}

The set Y is also finite set as it has fixed number of elements

#### 3. Infinite Sets

Set which has infinite elements that cannot be counted are infinite sets

Example 01
Set of number of people living in world
Y= { x : x is number of people living in world}

This set Y is infinite set as the number of people living in world is very huge number and very difficult to count as the number of people living in earth is changing continuously

Example 02
Set of natural numbers
Z = { x : x is number of natural number}
This is also infinite set as the number of natural number is infinite

#### 4. Equal Sets

Two sets are said to be equal if both contain same elements.

Example 01
Let there are two sets
A = {1, 2, 3, 4}
B = { 3, 1, 2, 4}
You can analyze that both the sets A and B contain the same elements, hence both are equal sets

Example 02
Check whether the set “ALLOY” and “LOYAL” are equal
Representing the sets of both the words:
N = { A, L, O, Y}
M = {L, O, Y, A}
Both the set N and M are equal as they contain same elements.

Example 03
Check whether the given sets are equal
A = {x : x-5=0}
B = {5}

The elements of set A is:
==> x-5 = 0
==> x = 5
Set A can be written as A = {5}
Both set A and B are equal as they contain same elements

### What is a subset?

Let there are two sets A & B
A = { 1, 2, 3}
B = { 1, 2, 3, 4, 5, 6}

You can observe that all elements of set A is also contained in set B
So we can say that A is subset of B ( A\subset B )

Definition 01
Set A is subset of set B if all the element of set A is also the element of set B
The symbol to represent subset is \subset

Definition 02 (Technical Definition)

Example 01
Let there are two sets A & B
A = { 1, 3, 5}
B = { x : x is odd number below 10}

Set B can be written as { 1, 3, 5, 7, 9}
So A is subset of B
Because all elements of set A is contained in set B

Example 02
Let there are two sets A & B
A = { a, e, i, o, u}
B = { a, b, c, d}
A is not subset of B as all the elements of A is not contained in set B
Also, B is not subset of A as all elements of B is not contained in set A

Note:
1. Empty set ( \phi ) do not contain any element.
So Empty set is subset of every set

2. Every set is the subset of itself
Set A contain all the elements of set A, hence set\quad A\subset Set\quad A

### Proper Subset and Superset

If A is subset of B, but B is not subset of A, then B is superset of A.

Example
Consider the two sets A & B
A = { 1, 2, 3, 4}
B = { 1, 2, 3, 4, 5, 6}
You can see that A is subset of B ( A\subset B)
B is not subset of A (as B contain more elements)
So B is superset of A and A is proper subset of B

### Set of Numbers in form of Intervals

Set can be used to represent the range of numbers
Suppose you want to represent the set containing number between 2 and 50, you will write
X = { y : 2< y < 50}
But this representation of range can also be done in the form of intervals

#### Open Interval

The above set X can also be represented in the form of interval as (2, 50)
The curved bracket () means open interval, i.e. the set is a range excluding 2 & 50
(2 , 50) = { y : 2 < y < 50}

#### Closed interval

If you want set containing range of number from 2 & 50 (including both 2 & 50)

The set can also be expressed in the form of closed bracket, i.e [2, 50]
The strong bracket [ ] means closed interval, i.e. the set of range 2 & 50 (including both 2 & 50)

#### Combination of Closed and Open interval

[2, 10) represent the set from 2 to 10; including 2 but excluding 10
(2, 10] represent the set form 2 to 10; excluding 2 but including 10

Hence if you want range of numbers, intervals can be used to represent the range.

#### Illustration of Intervals

a. Open Interval (a,b)
All the number between a & b, but excluding a & b

b. Closed Interval [a,b]
All number from a to b, including a & b

c. Closed and Open interval [a, b)
All number from a to b, including a & excluding b

d. Open and Closed interval (a, b]
All number from a to b, excluding a & including b

### What isPower Set

When you write all the subset of any set it became power set.

Example
X = { 1, 2, }
X is a set with elements 1, 2,

The possible subsets of X are
==> \phi
==> {1}
==> {2}
==> {1, 2} (Every set is a subset of itself)

Hence there are 4 possible subset of set X.
The set of all the subset is known as power set
Hence Y= { \phi , {1}, {2}, {1,2}} is a power set of set X

### What is Universal Set?

Universal set is the one which contain all the possible element of the set

Example 01
If you are forming set of population of different countries, then the population of the world is a universal set as it contains all the elements

Example 02
If you are studying numbers like prime numbers, even numbers, odd numbers then the set of natural number is a universal set as it contain all the elements

### Questions

(01) write the following sets in roster form
(a) A = {x : x is an integer and –3 ≤ x < 7}
(b) B = {x : x is a natural number less than 6}
(c) C = {x : x is a prime number which is divisor of 60}

Solution
(a) A = {x : x is an integer and –3 ≤ x < 7}
A = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6)

(b) B = {x : x is a natural number less than 6}
B = {1, 2, 3, 4, 5}

(c) C = {x : x is a prime number which is divisor of 60}
Divisor of 60 = 2 * 2 * 3 * 5
Prime number divisors are 2, 3, 5
C = { 2, 3, 5}

(02) List all elements in the sets
(a) A= { x : s is a letter in word ” LOYAL”}
(b) B = {x : x is a month of a year not having 31 days}

Solution
(a) A = (“L”, “O”, “Y”, “A”)
(b) B = (February, April, June, September, November)

(03) Write the following in set builder form
(a) {3, 6, 9, 12}
(b) {5, 25, 125, 625}

Solution
(a) {3, 6, 9, 12}

(b) {5, 25, 125, 625}

(04) In the following, state whether A=B or not
(i) A ={ a, b, c, d}
B = { d, c, b, a}

(ii) A ={ 4, 8, 12, 16}
B = {8, 4, 16, 18}

(iii) A= { 2, 4, 6, 8, 10 }
B = { x : x is a positive even integer and x<=10}

Solution
(i) A = { a, b, c, d}
B = { d, c, b, a}
Both the set contain same element
Hence A = B

(ii) A = { 4, 8, 12, 16}
B = { 8, 4, 16, 18}
Observe that element 12 in set A is not present in set B
Hence A is not equal to B

(iii) A= { 2, 4, 6, 8 ,10}
B = { 2, 4, 6, 8, 10}
Both set contain same element
Hence , A = B

(05)Write down all the subsets of the following sets
(i) {a, b}
(ii) {1, 2, 3}

Solution
(i) Subsets of {a, b} are:
==> ( \phi ), {a} , {b} , {a, b}

(ii) Subsets of {1, 2, 3} are:
==> ( \phi ), {1}, {2}, {3}, {1, 2}, {2, 3}, {1,3}, {1, 2, 3}

(06) Write the following interval
(i) {x : x ∈ R, – 4 < x ≤ 6}
(ii) {x : x ∈ R, 0 ≤ x < 7}

Solution
(i) {x : x ∈ R, – 4 < x ≤ 6}
{ -3, -2, -1, 0, 1, 2, 3, 4, 5, 6}

(ii) {x : x ∈ R, 0 ≤ x < 7}
{0, 1, 2, 3, 4, 5, 6}

(07) A = {1, 2, {3, 4} , 5}
Which of the following statement is incorrect?

(i) {3, 4} ⊂ A
(ii) {3, 4} ∈ A
(iii) {{3, 4}} ⊂ A
(iv) 1 ∈ A
(v) {1, 2, 5} ⊂ A

Solution
(i) {3, 4} ⊂ A
Its asking if {3, 4} is subset of set A
False because 1, 2, {3, 4}, 5 are elements of A
{{3, 4}} is subset of A

(ii) {3, 4} ∈ A
Correct
The element of A are 1, 2, {3, 4}, 5

(iii) {{3, 4}} ⊂ A
Correct because of the reason explained in (i)

(iv) 1 ∈ A
1 belongs to A?
Correct, as elements of A are 1, 2, {3,4}, 5

(v) {1, 2, 5} ⊂ A
{1, 2, 5 } is subset of set A?
Correct because all the element 1, 2, 5 is present in set A