The collection of distinct objects is called Sets. Sets can be classified into different types of sets and some of them are explained in the below sections taking a few examples. Get Definitions of various types of sets in the following sections. To gain conceptual knowledge on this you can look at Set Theory and clear your queries regarding the concept if any. See the Example Problems on Sets and learn which one falls under what kind of set.
Empty or Null or Void Set
Any Set that doesn’t have any elements in it is called an Empty Set. It is also called as Void or Null Set. The Symbol used to denote the Empty Set is {} or φ.
Example:
1. A = {x : 7 < x < 8, x is a natural number}
The Given Set will be a Null Set because there is No natural number between numbers 7 and 8.
2. Q = {z: z is a whole number which is not a natural number,z ≠ 0}
0 is the only whole number that is not a natural number. Since z ≠ 0 there is no other possibility thus the Set Q is {} or φ.
Singleton Set
If a Set has only one element then it is called a Singleton Set.
Example:
1. A = {x : x is neither prime nor composite}
Set A is Singleton Set since it has only one element i.e. 1 as it is neither prime nor composite.
2. B = {y : y ∈ N and y² = 9}
Set B is Singleton Set since there is only one element whose square is 9 and it is 3.
3. B = {z : z is a even prime number}
Set B is Singleton Set as 2 is the only even prime number.
Finite Set
Any Set that is either empty or contains a finite number of elements i.e. countable elements is called a Finite Set.
Example
S = { x | x ∈ N and 50 > x > 30 } is finite since the number of elements that satisfies the condition given are countable.
B = {a, e, i, o, u} is finite since it represents the vowel letters in the English Alphabet.
Infinite Set
Contrary to the Finite Set the number of elements in the Set is infinite then it is Called an Infinite Set.
Example
A = {y: y is a point on a line} is an infinite set as there will be an infinite number of points on a line.
A = {x: x is a real number} is an infinite set because there will be an infinite number of real numbers.
Equal and Unequal Sets
Two Sets A and B are said to be equal if they have the same elements irrespective of the order of the elements in the Set. Equal Sets are denoted by A = B whereas Unequal Sets are denoted by A ≠ B.
Example:
If A = { a, e, i, o, u} B = { i, e, u, o, a} then A = B since both the Sets have Same Elements no matter what the order is.
C ={ 4, 6, 7, 8} D = { 4, 5, 7,9 } then C ≠ D as it has different elements.
A = {x, y, z} and B = {u, v, x, y, z} in this case also A = B because both the Sets have same elements.
Equivalent Set
Equivalent Sets are those which contain the same number of elements irrespective of the elements within them.
A = {1, 2, 3, 4, 5} and B = {8, 9, 10, 11, 12} are equivalent sets because both these sets have 5 elements each.
S = {13,23, 33 …} and T = {y : y3 ϵ Natural number} are also equivalent sets.
Universal Set
Universal Set is the base for all other sets found. However, this depends on the context and can be either finite or infinite. All other sets are subsets of Universal Set and Universal Set is represented by U.
Example:
Real Numbers is a Universal Set for all Whole numbers, natural, odd and even rational, irrational numbers.
Power Set
Before heading into the concept of Power Sets you need to be aware of Sub Sets. Power Set is nothing but the Collection of all the Subsets. If a set has n elements the total number of subsets that can be formed are 2n.
Example:
If Set A = {-5,11,8}
The Power Set of A is P(A) = {ϕ, {-5}, {11}, {8}, {-5,11}, {11,8}, {8,-5}, {-5,11,8}}
Cardinal Number of a Set
The number of distinct elements present in the Set is called Cardinal Number of a Set. Consider a Set A and its Cardinal Number is represented as n(A).
Example:
A = Set of Letters in the Word Mathematics
The distinct letters in the given word are as such
A = { M A T H E I C S}
n(A) = 7