That definition is a (rare) example of Rudin doing things inefficiently. He could have defined 
for each non-negative integer 
to be the set of non-negative integers less than 
, so that 
, 
, 
, etc. Then he could have defined a set 
to be finite if and only if 
for some 
(where 
includes 
). This is essentially the usual set-theoretic definition stripped of some set-theoretic detail that would be out of place here.
Top answer 1 of 4
17
That definition is a (rare) example of Rudin doing things inefficiently. He could have defined for each non-negative integer to be the set of non-negative integers less than , so that , , , etc. Then he could have defined a set to be finite if and only if for some (where includes ). This is essentially the usual set-theoretic definition stripped of some set-theoretic detail that would be out of place here.
2 of 4
3
The parenthetical remark is just saying that formally the definition of finite set does not apply to the empty set, but the empty set is taken to be finite by convention.
If this bothers you, note it is possible to define a set as infinite precisely when there exists a proper subset of it with a bijection to the set. This captures in one go what it means to be infinite.
Wolfram MathWorld
mathworld.wolfram.com › FiniteSet.html
Finite Set -- from Wolfram MathWorld
September 12, 2002 - The empty set is also considered as a finite set, and its cardinal number is 0. A finite set can also be characterized as a set which is not infinite, i.e., as a set which is not equipollent to any of its proper subsets.
Videos
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.
vedantu.com
vedantu.com › maths › finite and infinite sets: definitions, properties & examples
Finite and Infinite Sets: Key Concepts & Examples Explained
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 infini
vedantu.com
vedantu.com › maths › finite and infinite sets: definitions, properties & examples
Finite and Infinite Sets: Key Concepts & Examples Explained
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.
vedantu.com
vedantu.com › maths › finite and infinite sets: definitions, properties & examples
Finite and Infinite Sets: Key Concepts & Examples Explained
Reddit
reddit.com › r/explainlikeimfive › eli5: why is empty set considered to be finite?
r/explainlikeimfive on Reddit: ELI5: Why is empty set considered to be finite?
July 1, 2016 -
I googled my question, but I'm afraid that I still don't get it.
Top answer 1 of 3
4
When you say a set is "infinite" or "finite" we are usually talking about its cardinality that is, how many members are in the set. The empty set has 0 members, and 0 is a finite number, so the empty set is finite. By contrast, the set of all integers has infinite members (its cardinality is infinity) which is, obviously, not finite.
2 of 3
1
Because it isn't infinite? It has a specific number of elements: zero elements. I don't know why you would think it isn't finite?
Cuemath
cuemath.com › algebra › finite-and-infinite-sets
Finite Sets and Infinite Sets - Definition, Difference, Properties, Examples
The number of elements in an empty set is definite, that is, zero, therefore, it is a finite set. The cardinality of a finite set is the number of members or elements present in the set. For example, set A is a set of all English alphabets, is a finite set. The cardinality of the set of English ...
VEDANTU
vedantu.com › maths › finite and infinite sets: definitions, properties & examples
Finite and Infinite Sets: Key Concepts & Examples Explained
December 11, 2020 - 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. A set which is not a finite set is infinite.
Quora
quora.com › Why-does-an-empty-set-consider-a-finite-set-Why-is-zero-considered-a-finite-number
Why does an empty set consider a finite set? Why is zero considered a finite number? - Quora
Answer (1 of 5): a number is finite if its absolute value is less than some natural number, and |0|
BYJUS
byjus.com › maths › finite-and-infinite-sets
Finite and Infinite sets
So, with a cardinality of zero, an empty set is a finite set. If a set is not finite, it is called an infinite set because the number of elements in that set is not countable, and also we cannot represent it in Roster form.
Published October 27, 2022 Views 31K
Quora
quora.com › Is-an-empty-set-finite
Is an empty set finite? - Quora
Answer (1 of 4): Yes. n(\emptyset) = 0 where n(\emptyset) is the Cardinality of the set. To view the \LaTeX source click '\cdots' \to Suggest Edits \LaTeX composed at Formula Sheet
askIITians
askiitians.com › iit-jee-algebra › set-relations-functions › set-theory › finite-and-infinite-sets
Finite and Infinite Sets - Study Material for IIT JEE | askIITians
The empty set is a finite set with a cardinality of zero. A set which is not a finite set is called an Infinite Set. Or if you cannot count the number of elements of a particular set then it is said to be an infinite set.
That definition is a (rare) example of Rudin doing things inefficiently. He could have defined for each non-negative integer to be the set of non-negative integers less than , so that , , , etc. Then he could have defined a set to be finite if and only if for some (where includes ). This is essentially the usual set-theoretic definition stripped of some set-theoretic detail that would be out of place here.
Reddit
reddit.com › r/learnmath › what happens if you assume the empty set is infinite?
r/learnmath on Reddit: What happens if you assume the empty set is infinite?
November 1, 2020 -
This came up in a course that we always assume the empty set is finite, is there anyone who has worked out a more formal or structured proof of why the empty set is finite? Or better yet used some kind of proof by contradiction to show "if we assume the empty set is infinite, we get x contradiction"?
Thanks!
Top answer 1 of 7
4
We don't assume the empty set is finite. We define the word "finite" in a way which includes the empty set. For example, one obvious way to do this is to define "finite" to mean "cardinality which is a natural number", and |{}|=0, a natural number, therefore {} is finite. If you did not define "finite" this way, and instead defined terms in a way that excluded {}, then your terminology would simply not align with what we want words to mean. For example, you would not be able to say "an infinite set is one with a bijection to a proper subset of itself" (since {} does not have that property), nor would you be able to say "every infinite set has arbitrarily large finite subsets" (since {} does not).
2 of 7
3
What do you mean? By definition, the cardinality of the empty set is 0.
Wikipedia
en.wikipedia.org › wiki › Finite_set
Finite set - Wikipedia
January 28, 2026 - All finite sets are countable, but not all countable sets are finite. (Some authors, however, use "countable" to mean "countably infinite", so do not consider finite sets to be countable.)
Top answer 1 of 2
192
Thank you for the question!This statement is true, because an empty set is finite set. For example the set like {3,5,7} will be finite set with a set of three natural numbers. So a finite set contains set of natural numbers or non negative numbers. Since empty set has zero number, or null element so it will be considered finite set.Hope it helps.
2 of 2
14
Kiran says that"Empty set will you agree with him or not Explain
Wikipedia
en.wikipedia.org › wiki › Empty_set
Empty set - Wikipedia
February 4, 2026 - ... {\displaystyle X} are complements of each other, the empty set is also closed, making it a clopen set. Moreover, the empty set is compact by the fact that every finite set is compact.
ALLEN
allen.in › home › jee maths › empty set
Empty Set: Definition, Examples, and Its Power Set
October 29, 2024 - Since 0 is a finite number, the empty set is considered finite. In contrast, an infinite set contains an unbounded or limitless number of elements, such as the set of natural numbers {1,2,3,....}.
Shaomaprep
shaomaprep.com › 2023 › 06 › 20 › understanding-sets-exploring-finite-infinite-and-empty-sets
Understanding Sets: Exploring Finite, Infinite, and Empty Sets – Read It Yourself Educational Resources
June 30, 2023 - Now that we understand the different types of sets, let’s explore how to distinguish between them. One way to classify sets is by their characteristics: Finite sets have a specific and countable number of elements. Infinite sets continue ...
BYJUS
byjus.com › maths › empty-set
Empty Set
Based on the number of elements in a set, we can define finite and infinite sets. However, we have another particular type of set called the empty set. The empty set is the unique set having no elements such that its cardinality is 0.
Published May 18, 2022 Views 31K
Filo
askfilo.com › cbse › smart solutions › 39) empty set is - (a) infinite set (b) finite set (d) unknown
39) Empty set is - (a) infinite set (b) finite set (d) Unknown set (C) Un..
November 2, 2024 - It is considered a finite set because it has a finite number of elements, specifically zero elements. Therefore, the correct answer is (b) finite set. Define the empty set: The empty set, denoted by ∅ or {}, is a set with no elements.
EduRev
edurev.in › act exam › act questions › empty set is a _______.a)infinite setb)finite...
Empty set is a _______.a)Infinite setb)Finite setc)Unknown setd)Universal setCorrect answer is option 'B'. Can you explain this answer? | EduRev ACT Question
Empty set is a Finite setAn empty set, also known as a null set, is a set that does not contain any elements. It is denoted by the symbol ∅ or {}. Despite not having any elements, an empty set is still considered a set in mathematics.Finite ...
Testbook
testbook.com › home › maths › finite and infinite sets
Finite and Infinite Sets: Definition, Venn Diagram & Examples
While it may seem counterintuitive, the empty set satisfies the criteria of being finite because it has zero elements.We can say empty sets are finite sets with a cardinality of zero.
CK-12 Foundation
ck12.org › all subjects › cbse math › sets and its types › can a finite set be empty?
Flexi answers - Can a Finite Set be empty? | CK-12 Foundation
September 11, 2025 - Yes, a finite set can be empty. This is known as the empty set, or the null set, and it contains no elements. In set notation, it is represented as @$\begin{align*}\{\}\end{align*}@$ or @$\begin{align*}\emptyset\end{align*}@$.