Definitions of Finite Set and Infinite Set
A finite set in mathematics is a set that has a finite number of elements. In simple words, it is a set that you can finish counting. For example, {1,3,5,7} is a finite set with four elements. The element in the finite set is a natural number, i.e. non-negative integer. A set S is called finite if there exists a bijection f:S = {1,……,n} for natural number n. The empty set {} is also considered finite. So, S is a finite set, if S admits a bijection to some set of natural numbers of the form {|x| < n}.
An infinite set is a set with an uncountable number of elements. We use dots to represent the infinite elements in a roaster. For example, a set of infinite natural numbers. {1,2,3,4,...}.
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The Cardinality of Finite Set
The cardinality of a finite set is n(A) = a, here a represents the number of elements of set A.
Whereas, the cardinality of the set A of all English Alphabets is 26, as the number of elements (alphabets) is 26. So, n(A) = 26.
It shows that you can list all the elements of a finite set and write them in curly braces or the form of Roster. Sometimes, the number of factors may be too big, but somehow it is countable or has a starting and ending point. Then this type of set is called a Non-Empty Finite Set. The number of elements is denoted with n(A) and if n(A) is a natural number then only it is a finite set.
Is an Empty Set a Finite Set?
An empty set is a set which has no elements in it. It is represented as { }, which shows that there is no element in the given set. The cardinality of an empty set is 0 (zero) as the number of elements is zero.
A={ } or n(A)=0.
The finite set is a set with countable elements. As the empty set has zero elements in it, so it has a definite number of elements.
Therefore, an empty set is a finite set with cardinality zero.
What is the Infinite Set?
A set which is not a finite set is infinite. If the number of elements is uncountable, then also it is called an infinite set. Unlike finite sets, we cannot represent an infinite set in roster form easily as its elements are not limited. So, dots are used to describe the infinity of the set.
What will you learn in the Finite and Infinite Sets Chapter?
In this particular chapter, you come to know about the basic definitions of the finite set and infinite sets. Further in the topic, Vedantu maths experts have explained the properties of the sets.
Properties of Finite sets
Following are the conditions of the finite sets.
Two subsets always form a subset.
The power set of finite sets is finite.
A subset of finite sets is finite.
Properties of Infinite sets
Following are the conditions of the infinite sets.
Union of two infinite sets is an infinite set.
The power set of infinite sets is infinite.
The superset of the infinite set is an infinite set.
Graphical Representation of Finite and Infinite Sets
Also, Vedantu maths teachers have provided you with a graphical representation of the sets that clear the unwanted confusion about finite sets.
In addition to that, Vedantu has brought a good amount of solved examples to help you understand these topics thoroughly and expose you to higher-order thinking skills (HOTS) questions.
Solved Example
Q1. Which of the following sets are finite or infinite?
The set of months of a year.
{1,3,5,…..}.
Answer: 1. The set of months can be represented as A= {Jan, Feb, Mar, Apr, May, Jul, Aug, Sep, Oct, Nov, Dec}. It forms a set of countable elements with the number of elements =12. Hence, it is a finite set.
Answer: 2. The set {1,3,5,…} has all the natural numbers but does not consist of any ending point. This makes it an uncountable set, and so it is an infinite set.
Q2. What is the Cardinality of Infinite Sets?
Answer: Cardinality of a set is expressed as n(A) = x, where x is the number of elements in the set A.
The number of elements in an infinite set is unlimited, so the cardinality of the infinite set n(A) = infinity.
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FAQs on Finite and Infinite Sets
1. What is the main difference between a finite and an infinite set?
The main difference lies in the number of elements they contain. A finite set has a definite, countable number of elements. You can, in principle, list all its elements and come to an end. An infinite set has an endless or uncountable number of elements, meaning the list of its elements goes on forever.
2. How can you determine if a given set is finite or infinite?
To determine if a set is finite or infinite, check if its elements can be counted up to a final number. A key method is to see if the process of counting its distinct elements has an end.
A set is finite if the counting process eventually stops. For example, the set of students in your class.
A set is infinite if the counting process never ends. For example, the set of all whole numbers {0, 1, 2, 3, ...}.
As per the CBSE syllabus, if a set can be put into a one-to-one correspondence with a subset of natural numbers {1, 2, 3, ..., n} for some natural number n, it is finite. Otherwise, it is infinite.
3. Can you provide some examples of finite and infinite sets from daily life?
Certainly. Here are some real-world examples:
Finite Set Examples: The set of days in a week {Monday, Tuesday, ..., Sunday}, the set of pages in a specific book, or the set of all letters in the English alphabet.
Infinite Set Examples: The set of all points on a straight line, the set of all numbers you can think of, or the set of all concentric circles that can be drawn from a single point.
4. What are the key properties of finite sets according to set theory?
Finite sets have several distinct properties that are important in set theory:
The subset of any finite set is always finite.
The union of two (or any finite number of) finite sets results in another finite set.
The power set (the set of all possible subsets) of a finite set is also finite. If a finite set has 'n' elements, its power set will have 2n elements.
5. Is an empty set (null set) considered finite or infinite, and why?
An empty set, denoted as {} or ∅, is considered a finite set. The reason is that a finite set is defined as one whose elements can be counted. The number of elements in an empty set is zero (0), which is a definite, countable number. Therefore, its cardinality is 0, which confirms its status as a finite set.
6. What is the importance of 'cardinality' in distinguishing between finite and infinite sets?
The cardinality of a set is the total number of distinct elements within that set; it essentially measures the "size" of the set. Its importance is fundamental in this topic:
For a finite set, the cardinality is a specific non-negative integer (e.g., 0, 1, 5, 100). For the set A = {a, e, i, o, u}, the cardinality n(A) = 5.
For an infinite set, the cardinality is not a finite number. We say its cardinality is infinite. The set of natural numbers N = {1, 2, 3, ...} has an infinite cardinality.
7. Why is the set of all natural numbers considered infinite, even though it starts at 1?
The set of all natural numbers N = {1, 2, 3, 4, ...} is considered infinite because it has no last element or upper bound. While it has a clear starting point (1), you can always add 1 to any given natural number to get another, larger natural number. This process can be continued indefinitely, meaning there is no "end" to the set, which is the defining characteristic of an infinite set.
8. Is the set of all sand grains on Earth finite or infinite?
This is a classic example to understand the mathematical definition. The set of all sand grains on Earth is finite. Although the number is astronomically large and practically impossible to count, it is still a definite, fixed quantity. In mathematics, 'infinite' refers to something that is truly without any bound, which is different from being just very, very large.
9. Why is an interval of real numbers, like [0, 1], considered an infinite set?
An interval of real numbers like [0, 1] is an infinite set because between any two distinct numbers in that interval, you can always find another number. For example, between 0.1 and 0.2, you can find 0.11, 0.12, 0.111, and so on. This process of finding a new number between two existing ones can be repeated endlessly. This means there is an uncountable number of elements packed into that small segment of the number line, making it a classic example of an infinite set.
10. If we cannot count the elements of infinite sets, how can they be compared?
This is an advanced concept in set theory. Infinite sets are compared using one-to-one correspondence. If you can pair up every element from one set with a unique element from another set, with no elements left over in either, the sets are said to have the same cardinality (size). For example, the set of all even numbers {2, 4, 6, ...} can be perfectly paired with the set of all natural numbers {1, 2, 3, ...}. This surprising result shows that some infinite sets can have the same 'size' even if one is a proper subset of the other, leading to the idea of different types of infinity.
