What Are Mathematical Sets and Venn Diagrams?

Learn about elements, set-builder notation, intersecting sets, and Venn diagrams

The Fibonacci Sequence

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Mathematically, a set is a collection or list of objects. Sets are not just comprised of numbers but can contain anything including the food in your refrigerator, the planets in the solar system, or ships named Enterprise.

Examples of Sets

Even though sets can contain anything, sets often refer to numbers that fit a pattern or are related in some way, such as:

  • The set of positive even numbers less than 10, for example, is 0, 2, 4, 6, 8
  • The set of ​factors for the number 12, for example, is 1, 2, 3, 4, 6, 12
  • The set of prime numbers less than 50, for example, is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Set Notation

The objects in a set are called elements, and sets use the following notation or conventions:

  • Single, uppercase letters such as J, E, or F identify sets
  • Lowercase letters or numbers are elements of a set
  • Curly braces { } denote a list of elements in a set
  • Commas separate set elements

So, examples of set notation would be:

  • J = {jupiter, saturn, uranus, neptune}
  • E = {0, 2, 4, 6, 8}
  • F = {1, 2, 3, 4, 6, 12}

Element Order and Repetition

Elements in a set do not have to be in any particular order, so you could also write the set J above as:

J = {saturn, jupiter, neptune, uranus}


J = {neptune, jupiter, uranus, saturn}

Repeating elements doesn't change the set either, so:

J = {jupiter, saturn, uranus, neptune}


J = {jupiter, saturn, uranus, neptune, jupiter, saturn}

are the same set because both contain only four different elements: Jupiter, Saturn, Uranus, and Neptune.

Sets and Ellipses

If a set contains an infinite number of elements, an ellipsis (...) expresses that the pattern of the set continues on forever in that direction.

For example, the set of whole numbers starts at zero but has no end, so you could write it as:

​{0, 1, 2, 3, 4, 5, }

Another special set of numbers that has no end is the set of integers. Since integers can be positive or negative, however, the set uses ellipses at both ends to show that the set goes on forever in both directions:

{, −3, −2, −1, 0, 1, 2, 3, }

Another use for ellipses is to fill in the middle of a large set. You could write the set of even numbers between 0 and 100 as:

​{0, 2, 4, 6, 8, …, 94, 96, 98, 100}

The ellipsis shows that the pattern — even numbers only — continues through the unwritten section of the set.

Special Sets

Special sets that are used frequently are identified using specific letters or symbols. These include:

  • Ø or { } — the empty set — a set containing no elements
  •  — the universal set — a set containing all elements relative to a particular set definition
  • Z — the set of all integers where Z = {, −3, −2, −1, 0, 1, 2, 3, }
  • N — natural numbers (positive integers) where N = {1, 2, 3, 4, 5, }

Roster vs. Descriptive Methods

Writing out or listing the elements of a set, such as the set of the inner or terrestrial planets in our solar system, is called roster notation or the roster method.

T = {mercury, venus, earth, mars}

Another option to identify the elements of a set is to use the descriptive method, which uses a short statement or name to describe the set such as:

T = {the terrestrial planets}

Set-Builder Notation

An alternative to the roster and descriptive methods is to use set-builder notation, which is a shorthand method describing the rule that the elements of the set follow (the rule that makes them members of a particular set).

Set-builder notation for the set of natural numbers greater than zero is:

{x | x ∈ N, x > 0}


{x : x ∈ N, x > 0}

In set-builder notation, the letter "x" is a placeholder which you can replace with any other letter.

Shorthand Characters

Shorthand characters that are used with set-builder notation include:

  • The vertical bar or colon (| or : characters) are separators read as "such that"
  • The lowercase epsilon ( character) is read as "is an element of"
  • The  character is read as "not an element of"

So, you would read {x | x ∈ N, x > 0} as, "The set of all x, such that x is an element of the set of natural numbers, and x is greater than 0."

Sets and Venn Diagrams

A Venn (or set) diagram shows relationships between the elements of different sets. The overlapping section of a Venn diagram shows the intersection of two or more sets (elements common to both sets).

Venn Diagram of an Intersection Set

The set-builder notation for the operation (the upside-down "U" means "intersection") is:

E ∩ F = { x | x ∈ E x ∈ F }

The rectangular border and the letter U in the corner of the Venn diagram above represent the universal set of all elements under consideration for this operation:

U = { 0, 1, 2, 3, 4, 6, 8, 12}
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