In this chapter we will learn different types of sets with properties & examples.
To understand the concept you should have basic knowledge of sets and its representation.
Different Types of Sets
Given below are some of the types of sets with examples.
Null Set
A set which do not contain any element is known as null set.
⟹ Null set is also called Empty set or Void set.
⟹ It is represented by symbol “𝜙” (known as “Phi”)
⟹ In roster form null set is represented as { }
Note that there is no element inside the curly bracket.
Examples of Empty Set
Example 01
A = { x : 2< x < 3 & x 𝜖 N }
Explanation:
Given above is the set with name A.
The set contains element x such that;
⟹ x lies between 2 and 3, and;
⟹ x is a natural number
Since there is no natural number between 2 & 3, the above set A is an empty set.
i.e. A = { }
Example 02
B = { x : x > 2 & x is even prime number }
Explanation
Given above is the set with name B.
Set B contain element x such that;
⟹ x is greater than 2
⟹ x is even prime number
There is no even prime number greater than 2.
Hence set B is a null set.
Example 03
Check if the below set is null set or not?
A = { 0 }
Explanation
The above set contains number 0 in it.
Since there is one element present, it is not a null set.
Singleton Set
A set containing one element is known as Singleton set.
⟹ The set contains exactly one element, not less not more.
Examples of Singleton Set
Example 01
Y = { 9 }
Explanation
Given above is a set named Y containing one element 9.
Since the set contains one element, it is a singleton set.
Example 02
Y = { x : x is a even prime number}
Explanation
Given above is the set named Y.
Set Y contains element x such that;
⟹ x is a even prime number
Since 2 is the only even prime number in the number system, the set can be written as:
Y = { 2 }
Hence, set Y is a singleton set.
Example 03
M = { x : x < 2 & x 𝜖 N }
Explanation
Given above is the set with name M.
Set M contain element x such that;
⟹ x is less than 2
⟹ x belongs to natural number
Since 1 is the only natural number which is less than 2, the set can be written as:
M = { 1 }
There is only one element in the set M.
Hence it is a singleton set.
Finite Set
A set whose elements can be easily counted is called Finite set
⟹ the listing of elements in finite set is limited and terminates at certain countable number.
⟹ the empty set is also a type of finite set.
⟹ the elements of finite set can be easily expressed in roster form
Examples of Finite set
Example 01
A = { 5, 7, 9, 10, 15, 26, 35 }
Explanation
The above set A is a finite set since it contain limited elements.
There are total of 7 elements in set A.
Example 02
A = { Set of Letters in English Alphabets }
Explanation
Set A is a finite set as we know there are 26 letter in English Alphabets.
Since the number of alphabet is ending and easily countable, the set is finite.
Example 03
A = { Number of people in Football stadium }
Explanation
Set A is a finite set since the people are limited and can be counted easily.
Example 04
M = { x : x < 5 & x 𝜖 W }
Solution
Given above is the set named M.
The set M contains element x such that;
⟹ x is less than 5
⟹ x belongs to whole numbers
The set can be written as:
M = { 0, 1, 2, 3, 4 }
Hence, set M is a finite set.
Infinite Set
A set which contains uncountable and never ending elements is called infinite set.
⟹ Infinite sets contain unending elements
⟹ In infinite sets, the elements cannot be listed by natural numbers.
Since, there are infinite elements, the listing will go on and on and cannot be counted.
⟹ Infinite sets are difficult to represent in roster form.
Generally set builder form of representation is used for infinite sets.
Examples of Infinite set
Example 01
A = { Set of positive even numbers}
Explanation
We know that there are infinite number of positive even numbers.
Some examples of +ve even numbers are:
2, 4, 6, 8, 10, 12 . . . . . . . .
The numbers will go on and on.
Since the set contains unending elements, it is an infinite set.
Example 02
X = { Number of stars in sky }
Explanation
There are uncountable stars in the sky.
Hence, the set X is infinite set.
Example 03
Y = { x : x > 15 & x 𝜖 N }
Explanation
Given above is the set named Y.
Set Y contains element x such that;
⟹ x is greater than 5, and;
⟹ x belongs to natural number N
So the possible elements in set Y are;
Y = { 16, 17, 18, 19, . . . . .}
There are uncountable elements in the set Y.
Hence, set Y is an infinite set.
Equivalent Sets
Two sets are said to be equivalent if their cardinal numbers are same.
Cardinal Numbers is the number of unique element present in the given set.
⟹ Equivalent sets have same number of unique elements.
⟹ The elements can be different. The total number of unique element must be same.
⟹ If sets A & B are equivalent sets then; n (A) = n(B)
i.e. number of element in A = number of element in B.
Examples of Equivalent Sets
Example 01
A = { 3, 4, 7 }
B = { 9, 15, 33 }
Both sets A & B are equivalent sets.
Explanation
Set A contains three elements.
n (A) = 3
Set B also contains 3 elements.
n (B) = 3
We get, n(A) = n(B)
Hence, the sets A & B are equivalent sets.
Example 02
Y = { x : 3<x<10 & x 𝜖 N }
Z = { a : 12<a<19 & a 𝜖 N }
Both sets Y & Z are equivalent sets.
Explanation
Y = { x : 3<x<10 & x 𝜖 N }
Given above is set Y which contains element x such that;
⟹ x lies between 3 and 10
⟹ x belongs to natural number N
So the elements of set Y are:
Y = { 4, 5, 6, 7, 8, 9}
There are total of 6 elements.
n(Y) = 6
Z = { a : 12<a<19 & a 𝜖 N }
This is another set Z which contains element a such that;
⟹ a lies between 12 and 19
⟹ a belongs to natural number N
Elements of set Z are;
Z = {13, 14, 15, 16, 17, 18}
There are 6 elements in set Z.
n(Z) = 6
Since, n(Y) = n(Z) = 6, both the sets are equivalent sets.
Equal Sets
Two sets are said to be equal if both the sets contain same elements.
⟹ If A & B are equal sets then;
All elements of set A is present in set B & vice-versa.
⟹ if set A & B are equal then we write, A = B
Example of Equal Sets
Example 01
A = { 6, 10, 13, 7 }
B = { 6, 7, 10, 13 }
The above sets A & B are equal sets.
Explanation:
Set A is made of element 6, 7, 10, 13.
Set B is also made of same elements 6, 7 , 10 , 13.
Hence, both the sets are equal (i.e. A = B )
Example 02
Set A = { 21, 19, 21, 19 }
Set B = { 19, 21 }
Both sets A & B are equal.
Explanation
Set A is made of elements 19 & 21.
Set B is also made of elements 19 & 21.
Since set A & B are made of same elements, they are said to be equal.