Sets are classified into two types: finite and infinite. The phrase ‘Finite’ implies that it is countable, but the word ‘Infinite’ implies either finite or uncountable.
Learn about Finite and Infinite Sets Properties, their definitions, attributes, and other aspects, as well as different examples and problems.
Finite Set
Sets with a countable number of members are called finite sets. Because they can be numbered, finite sets are also known as countable sets. If the set elements have a countable number of members, the operation will run out of elements to list.
Finite set examples: P = 0, 3, 6, 9,…, 99
Q = a: an is an integer (1 a 10).
A collection of all English Alphabets
Another instance of a finite set:
A year’s worth of months.
M is an abbreviation for January, February, March, April, May, June, July, August, September, October, November, and December.
12 = n (M)
Because the number of items is countable, it is a finite set.
Cardinality of Finite Set
If ‘a’ indicates the number of items in set A, then a finite set’s cardinality equals n(A) = a.
As a result, the Cardinality of set A of all English Alphabets is 26 because there are 26 members (alphabets).
As a result, n (A) = 26.
Similarly, the cardinality of a set containing the months in a year is 12.
So, in this manner, we may list all of the items of any finite set, either in curly brackets or in Roster form.
Finite set properties
The finite set requirements listed below are always finite.
⦁ A subset of a finite set
⦁ The intersection of two finite sets
⦁ A power set of a finite set
Few Examples:
P = {1, 2, 3, 4}
Q = {2, 4, 6, 8}
R = {2, 3)
Because the elements are finite and countable, all P, Q, and R are finite sets.
R P, i.e. R is a Subset of P since all of the components of R are present in P. As a result, a finite set’s subset is always finite.
P U Q is 1, 2, 3, 4, 6, 8, implying that the union of two sets is also finite.
The number of items in a power set is equal to 2n.
As a set P has four items, the number of elements in the power set of set P is 24 = 16. As a result, it demonstrates that the power set of a finite set is finite.
Non- Empty Finite set
It is a set in which the number of members is large or simply the beginning or endpoint is specified. As a result, we designate it by the number of members, n(A), and if n(A) is a natural integer, it is a finite set.
Example:
S = a collection of the number of individuals who live in India
The population of India is difficult to estimate, yet it is close to a natural amount. As a result, we can refer to it as a non-empty finite set.
If N is a collection of natural numbers. As a result, the cardinality of set N is n.
N = 1,2,3,…n
X = x1, x2,…., xn
Y = x: x1 N, 1 I n, where I is a positive integer between 1 and n.
Is it possible to assert that an empty set is a finite set?
First, let’s define an empty set.
An empty set is a set that contains no elements and may be expressed as and illustrates that it contains no items.
P = { } Or ∅
Because the finite set contains the empty set that has zero items, the number of elements is definite.
As a result, an empty set is termed as the finite set with a cardinality of zero.
What is the definition of Infinite Set?
If a set is not finite, it is referred to as an infinite set since the number of items in that set cannot be counted and cannot be represented in Roster form. As a result, infinite sets are frequently referred to as uncountable sets.
As a result, the elements of an infinite set are represented by three dots (ellipses), representing the set’s infinity.
Infinite Sets Examples
W=0, 1, 2, 3, 4,… is a set of all whole numbers.
A collection of all the points on a line
The collection of all integers
Cardinality of Infinite Set
A set’s cardinality is n (A) = x, where x is the number of elements in a set A. Because the number of items in an infinite set is infinite, its cardinality is n (A).
Infinite Set Properties
⦁ The union of two infinite sets has an unlimited number of solutions.
⦁ An infinite set’s power set is unlimited.
⦁ An infinite set’s superset is also infinite.
Basic Difference Between Finite and Infinite Sets:
Let’s look at the distinctions between Finite and Infinite sets:
The sets can only be equal if their elements are the same. Hence, a set can only be equal if its elements are comparable; however, the set is infinite if the components are not comparable.
| Factors | Finite sets | Infinite sets |
| Number of elements | Countable Elements | Uncountable Elements |
| Continuity | It has a start and end elements | It is endless from start or end. Both sides could have continuity |
| Cardinality | n is the number of elements in the set | the number of elements is uncountable |
| union | The union of two finite sets is finite | The union of two infinite sets is infinite |
| Power set | The power set of a finite set is also finite | The power set of an infinite set is infinite |
| Roster form | Can be easily represented in roster form | Can’t be represented in Roster form |
Conclusion
As we know, a set is finite if it has both a starting and an ending point, but it is infinite if it has no end from either one or both sides. The following points can be used to determine whether a set is finite or infinite:
⦁ An infinite set is infinite from start to finish, yet both sides might have continuity, unlike a finite set with both start and end elements.
⦁ If a set has an infinite number of elements, it is infinite; if the elements can be counted, it is finite.
