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**Contents (Click to go to that section):**

- Finite Set Definition
- Finite Set Notation
- Infinite Set Definition
- Infinite Set Notation
- Set Functions
- Open and Closed Sets
- Universal Set

## 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}

or

{.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 be a family of sets, e, = {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.

## 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).

## Open and Closed Sets

An **open set** contains its boundary; it is a generalization of an open interval. A **closed set** does *not *contain its boundary and is a generalization of a closed interval. In topology, a closed set is defined an one whose complement is open.

## Open Set / Closed Set Examples

Every open interval is also an open set ^{[1]}. For example, the interval (3, 5) is open because any x-value in the set will be between 3 and 5. In other words, if you choose a number very close to one of the boundaries (3 or 5), there will always be a set of numbers surrounding it that does not contain the boundary. Let’s say you choose 3.001. The numbers 3.00001 and 3.01 are:

- In the interval [3, 5],
- Surround 3.001,
- Do not contain the boundary.

You could continue choosing number ad infinitum and never reach the boundary. This leads to an alternative definition of an open set, which is in terms of distance. A set (a, b) is open if it contains all numbers “sufficiently close” to a and b ^{2}.

## Properties of Open Set / Closed Set

- The complement of an open set is closed. For example, [3, 5] is closed because its complement is two open sets

(-∞ 3) ∪ (5, ∞). - Every union of open sets (the smallest set that contains both sets) is open.
- Every finite intersection of open sets is open.

However, the fact that the complement of an open set is closed does not mean that “closed set” and “open set” are antonyms. Sets can be open, closed, both, or neither ^{[3]}.

*Note: A **complement **is all elements, from a universal set, that are *not *in the set of interest. For example, if your universal set is {1, 2, 3, 4} then the complement of {1, 2} is {3, 4}.

## Open and Closed Set: References

[1] Knapp, A. (2005). Basic Real Analysis. Birkhäuser Boston.

[2] A Short Introduction to Metric Spaces: Section 1: Open and Closed Sets. Retrieved August 4, 2021 from: https://math.hws.edu/eck/metric-spaces/open-and-closed-sets.html

[3] Lamb, E. (2013). Can a Closed Set Be Open? Can an Open Set Be Closed? When Math and Language Collide. Retrieved August 4, 2021 from: https://blogs.scientificamerican.com/roots-of-unity/can-a-closed-set-be-open-can-an-open-set-be-closed-when-math-and-language-collide/

## Universal Set

A **universal set**, denoted by * U*, has all elements of interest. Elements of a set are a collection of items, so you can also think of the universal set as the overall collection of items you’re interested in: If you’re interested in studying freshmen students, then “all freshmen” is you’re universal set; if you’re studying the ethnic make up of political parties, then

*= all political parties. Anything you think of can be defined within a universal sets, from frogs to footballs, from numbers to electrons.*

**U****All other sets are subsets of the universal set**. For example, a list of the 52 U.S. states is a universal set, with subsets of Eastern states, Western states, states that begin with the letter A, and so on.

Defining ** U** is helpful for establishing a frame of reference for set problems [1]. The rule for a set is that each member of

*has to be clearly in the set, or not in the set. For example, you probably would not have*

**U***as a list of nice chocolate manufacturers, as there’s some ambiguity as to what “nice” is.*

*U*## Does the Universal Set Contain Everything?

The idea of the universal set has been around from before the 20th century, when mathematicians and philosophers first imagined a collection of all possible entities [2]. However, some set theories do not allow * U* to contain everything; Cantor and Bertrand Russell proved that

**cannot contain**

*U**everything*as it leads to paradoxes and inconsistencies. Other set theories (such as Zermelo–Fraenkel set theory) simply do not include

*at all.*

**U**## Venn Diagram of the Universal Set

The following Venn diagram shows the universal set with a subset A, a subset of interest (left) and with A as a subset of B, and both are subsets of * U* (right):

## Universal Set: References

[1] Wooland. Part 1 Module. Set Mathematics Sets, Elements, Subsets. Article posted on Florida State University website.

[2] https://home.cs.colorado.edu/~yuvo9296/courses/csci2824/sect14-sets1.html

## References

Grossman, C. (2010). Comparing Infinite Sets. Retrieved May 15, 2019 from: http://ime.math.arizona.edu/ati/Math%20Projects/C1_MathFinal_Grossman.pdf

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: http://www.math.tamu.edu/~mayaj/Chapter6_Sec6.2_f17completed505.pdf

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 http://faculty.kutztown.edu/wong/17FaMAT260/17FaMAT260Lecture03.pdf on January 4, 2018.

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