Before delving into the different components of finite and infinite sets components, take a quick look at the Finite and Infinite Sets definition.
What are finite sets?
A set having a limited number of elements is mathematically termed a finite set. Placing it the other way round, we can also say that a finite set is a combination of sets that allows one to start and finish the counting more efficiently.
The below example suggests a finite set, {2,4,6,8,10} which has five elements in it. Also known as the cardinality of a set, the number of elements comprising a finite set are natural numbers or non-negative integers.
The importance of finite sets in mathematics, physics, and various other branches related to science is significant. A peculiar significance to finite sets has been given in combinatorics which is the mathematical study of counting.
Many equations and assumptions are based on the pigeonhole principle, which suggests that a function that is injective from a larger finite set can never exist to a smaller finite set.
The existence of a junction causes a set to be finite, which is denoted by S.
For example, S { 1, n, }, where n is a natural number, representing the cardinality of the particular set denoted by S. Sets having the symbol as {} represents an empty set or is considered as a finite set having cardinality zero.
In infinite sets, the elements are written as ( a1, a2, a3…. an). These sets are always written in sequence.
A finite set in combinatorics with n elements is called an n-set and a subset consisting of the k elements. This element is called a k-subset. Let us check some other examples, {1,2,3 }; is 3 sets- it is a finite set with three elements, and {4,5} is a 2 subset, a finite set with two elements.
The basic properties of finite numbers
A finite set S having a proper finite subset has fewer elements than s itself, which results in the failure information of bijection between a finite S and a proper subset of S. Sets possessing these properties are called Dedekind-finite. With the use of ZFC axioms for set theory, every single Dedekind-finite set is always finite.
However, ZF does not prove this theory, which stands for Zermelo-Fraenkel axioms that do not have the axiom of choice. The axiom, which can be of a countable size and a weaker version of the same, is adequate to prove the equivalence.
It is termed surjection or a surjective function when any of the injective functions among the two finite sets of similar cardinality occur. Likewise, Injection is defined as any surjection among these two sets that are similar in several cardinalities.
Union of two finite sets is finite having,
S TS + T
Going by the principle of inclusion and exclusion
S T= S + T – S T
Usually, the union of any finite number of finite sets is finite. Cartesian product for the finite set is also finite with,
S x T= S x T
What are infinite sets?
The arrangement of numbers whose presence is intimated by the axiom of infinity is an infinite set. It is the only set that is directly required by the axioms to be infinite. Zermelo-Fraenkel’s theory of sets (ZFC) proves the existence of an infinite set.
However, it only shows that the existence of natural numbers follows it. An infinite set may also be termed as a set that is not finite as the number of elements in that set are not countable and cannot be represented in the roster form. Infinite sets are also known as uncountable sets.
Three dots or an ellipse represent the elements of an infinite set and the infinity of that set.
Examples of an infinite set
- All whole numbers who form a set, W= { 0,1,2,3…}
- A line containing a set of all points
- The set of all integers
Properties of infinite sets
The below-mentioned points are some of the properties of infinite sets.
- The union of a number of infinite sets is an infinite set.
- The power set of an infinite set is infinite.
- The superset of an infinite set is always infinite.
- The subset of an infinite set may or may not be infinite.
- Infinite sets can be counted or uncountable, taking the example of the set of real numbers that are not countable, but the set of integers is countable.
Some important points to remember with respect to finite and infinite sets are :
- An empty set with cardinality equal to zero is a finite set.
- The cardinality of rational numbers equals the cardinality of natural numbers.
- All finite sets are countable, whereas infinite sets may be or may not be countable.
The words Finite sets and infinite sets are opposite to each other. In simple words, ‘Finite’ means countable, whereas ‘Infinite’ stands for something we can count.
Solved Examples
Question 1:
Explain if the following sets are finite or infinite?
- i) set 1 = set of multiples of 10 less than 101
- ii) set of all integers.
Answer: set 1= set of multiples of 10; ess than 101 = { 10, 20, 30, 40, 50, 60,….,100 } is a finite set because the number of multiples of 10 less than 101 is finite.
- ii) set of all integers = it is an infinite set because there is an infinite number of elements in the set.
Question 2:
If set B = { …..,-5, -4, -3 }, determine whether the given set is a finite set or an infinite set ?
Answer: Set B = { …..,-5, -4, -3 }, is an infinite set as the elements of the set B start from a negative of infinity and therefore cannot be finite.
Now coming to the common difference between finite sets and infinite sets
Finite sets –
The finite sets can be counted, and the union of two finite sets is finite. A subset of a finite set is finite along with the power set of a finite set. Examples, set of even natural numbers less than 50 or the days of the week.
Infinite sets –
Infinite sets can be countable or uncountable; the union of two infinite sets is infinite. The subset of an infinite set may be finite or infinite. The power set of an infinite set is infinite. Common examples are real numbers.
Conclusion
Read this article thoroughly to have a clear idea regarding the finite and infinite sets and their distinctive characteristics.
Read More: Learn More About Linear Algebra With Different Types Of Matrices.
