Types of sets – Definition – Examples
Types of sets
In set theory, we will learn Type of sets as
Finite and Infinite sets
Singleton set
Cardinal number of a set
Equal and Equivalent sets
Empty set
Overlapping set
Subsets
Disjoint set
Universal set
1. Finite set:
Finite set: In a set number of elements is limited or numbers are counted is called a finite set.
Example: a. A = {2, 4, 6, 8, 10}
In above example set A of even numbers 2 to 10, the elements in set A is limited, so A is a finite set.
b. A team of football is a finite set, the players in team is a fix number.
2. Infinite set:
In a set number of elements is unlimited or counteless is called an infinite set.
Example: a. The set of whole numbers.
A = {0, 1, 2, 3, 4…}
The set of whole numbers is an infinite set, because the counting of numbers never ending the elements are countless.
b. The set of natural numbers.
A = {1, 2, 3, 4, 5…}
The set of natural numbers is an infinite set, because the counting of numbers never ending the elements are countless.
3. Singleton set:
In a set number of elements is only one is called a singleton set.
Example: a. A = {x : x is a whole number, x < 1}
a whole number less then 1 is 0, thus set A have only one element 0 and is a singleton set.
b. B = {x : x is a even prime number}
Here B is a singleton set because there is only one prime number which is i.e. 2.
4. Cardinal number of a set or Cardinality of a set:
Cardinal number of a set or Cardinality of a set is a measure of the number of elements of the set.
It is denoted by n(A)
n(A) is read as the number of elements in set A.
Example-
a. A = {2, 3, 4, 5…}
The Cardinality of a set A is 4.
It is denoted as n(A) = 4
b. A = {1, 6, 7, 9…}
The Cardinality of a set A is 4.
It is denoted as n(A) = 4
5. Equal sets:
Two sets A and B are said to be equal, if Set A and set B will have same elements.
In other words every elements of A is also an element of B and every element of B is also an element of A.
Example-
a. A = {1, 2, 3, 4…}
B = {4, 2, 3, 1…}
Therefore, A = B
6. Equivalent sets:
Two sets which have the same number of elements, or same cardinality are called equivalent sets.
Example: A = {1. 2. 3, 4, 5} and B = {x, y, z, l, m}
Since the two sets A and B contain the same number of elements 5, therefore they are equivalent sets.
7. Empty set:
A set which does not contain any element is called Empty set, or null set or void set.
Empty set is denoted by { } or ∅ (Greek letter phi).
Example
a. The set of natural numbers less than 0,
we know that there is no natural number less than 0.
Therefore, it is an empty set.
b. A = {x : x ∈ N, 6 < x < 7}
Here A is an empty set because there is no natural number between 6 and 7.
8. Overlapping sets:
Two sets that have at least one common element are called overlapping sets.
Example
A = {2, 3, 4} and B = {4, 5, 6}
The two sets A and B have an element 4 in common. Therefore they are called overlapping sets.
9. Subset:
A set A is said to be subset of set B, if every element of A is also an element of B.
If A is subset of B, in symbol we write A⊂B
Example: A = {2, 3, 4} and B = {2, 3, 4, 5, 6}
Here every element of a set A is also in set B, so
A⊂B.
10. Disjoint sets:
Two sets are disjoint, if they have no common element. Or Two sets A and B are said to be disjoint, if Set A and set B will have no common element.
In other words any element of A is not in B and any element of B is not in A.
Example-
a. A = {1, 2, 3, 4…}
B = {5, 6, 7, 8…}
Sets A and B are said to be disjoint because set A and set B will have no common element.
11. Universal set:
A universal set is a set which contains all the elements or objects of other sets. All other sets are subsets of the universal set.
Universal set is represented by capital letter “U”.
It’s all subsets are represented by other capital letters like A, B, C, D etc.
Example: A = {1, 2, 3,}, B = {3, 4, 5, 6, 7, 8}
Therefore, by the definition of universal set, it contains all the elements of other sets.
U = {1, 2, 3, 4, 5, 6, 7, 8},