Calculus How To

Finite Set and Infinite Sets; Set Function

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finite set

A set can contain anything from zero to infinity elements.

Contents (Click to go to that section):

  1. Finite Set Definition
  2. Finite Set Notation
  3. Infinite Set Definition
  4. Infinite Set Notation
  5. Set Functions

1. Finite Set Definition

A finite set has a certain, countable number of objects. For example, you might have a fruit bowl with ten pieces of fruit. More technically, a finite set has a first element, second element, and so on, until the set reaches its last element.

If you can count the number of objects in your set, that’s a finite set.

Now try counting the number of stars in the universe. You won’t be able to, because there are an infinite number of items in the set of all stars.

2. Finite Set Notation

In notation, a finite set is:
{1, 2, 3, 4, 5}
Where you can replace 1 through 5 with any amount of any number. For example:
{101, 222, 433, 97894, 5213457}
{.21, .22, .43, .7654, .975}

3. Infinite Set: Definition

If you can’t count the number of objects, it’s an infinite set.

More technically, infinite sets don’t have a last element (e.g. a last number, letter, or object); The last of a last element makes counting go toward infinity. “The number of stars in the universe” is an example of an infinite set.

4. Infinite Set: Notation

If you see three dots “…” at the end of a set (without any numbers of variables following the dots), that means it contains an infinite number of items.

Usually, but not always, the items in the infinite set will give you a clue to the missing contents. For example:

  • {1, 2, 3, 4, 5, …} indicates it goes on and on to 6, 7, 8, 9, 10 … and beyond (basically, keep counting and never stop).
  • {100, 200, 300, …} indicates you keep counting by one hundred until infinity.

Defining the Set Function

The idea of a set function is was first mentioned by Cauchy in 1841 (cited in Medvedev, 1991), although he didn’t actually call it a “set function”. The founders of set function theory are considered to be Peano and Lebesgue, but it wasn’t really developed until the early 20th century. Nathanson’s 1957 work Theory of Functions of Real Variables includes this simple definition:

“let set function be a family of sets, e, set function = {e}. If to each set…there corresponds some number…we say that a set function is defined on [that] family.”

It should be noted though, that this definition is probably over-simplistic, because it could (in theory) be reduced to a transformation, which leads to a mathematical paradox. Therefore, a transformation, although related to set functions, is something that isn’t usually included in the conversation.

Interesting, an integral is a type of set function, where the input is the area under a curve, and the output is one number.

An integral is a type of set function.

Set-to-Element Correspondence

The idea of a set function is in contrast to most of the functions you’ve probably come across in calculus so far. Typically, when we talk about a function being “one-to-one correspondence” or “many-to-one correspondence“, it’s in relation to inputs that are each a single value (usually on the x-axis). With set functions, the correspondence is set-to-element (Medvedev, 1991).


Grossman, C. (2010). Comparing Infinite Sets. Retrieved May 15, 2019 from:
Hersch, R. (1997). What Is Mathematics, Really? Oxford University Press.
Johnson, M. (2017). Section 6.2 The Number of Elements in a Finite Set. Retrieved May 15, 2019 from:
4.7 Cardinality and Countability
Medvedev, F. (1991). Scenes from the History of Real Functions (Science Networks. Historical Studies) 1991st Edition
Nathanson, I. (1957). Theory of Functions of Real Variables – IN RUSSIAN CYRILLIC .
Royden, H. L. and Fitzpatrick, P. M. Real Analysis. Pearson, 2010.
Wong. Linear Algebra: Real Vector Spaces. Retrieved from on January 4, 2018.

Stephanie Glen. "Finite Set and Infinite Sets; Set Function" From Calculus for the rest of us!

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