Singleton set

In this article we explain what is a singleton set and see examples and properties of these sets.

Table of Contents

What is a singleton set?

A singleton set is one that contains exactly one element. This element can be any object, number, symbol, or entity defined within the context of the set.

Examples

Below are some examples of singleton sets given by roster form or set builder notation.

Set A = {a} is a singleton set since it contains only one element: "a".
Set B = {1, 1, 1, 1, 1} is a singleton set because it has only one object, regardless of whether it is written multiple times.
Set C = {x | x is the initial letter of the word "house"} is a singleton set. Given by listing, C = {h}.
The set of solutions to the linear equation *2x+1=3* is a singleton set, as {x | *2x+1=3*} = {1}.
The power set of the empty set is a singleton set, as it has only one element, the empty set itself: P(Ø)={Ø}.
Set M = {x | x is an integer and 2 < x < 4} is a singleton set, as we can deduce that M = {3}.
Set T = {x | x is the planet inhabited by humans} is a singleton set, as T = {Earth}.
Set R = { {1,2,3} } is a singleton set, as its only element is the set {1,2,3}.
Set L = {x | x is a natural satellite of the Earth} is a singleton set, as the set can be expressed as M = {Moon}.
The intersection of sets P = {5, 7, 9} and Q = {0, 1, 9} is P ∩ Q = {9}, a singleton set.

Venn diagram of a singleton set with repeated elements. — Venn diagram of the singleton set from example 2

Properties

Singleton sets have the following properties.

A singleton set has only two subsets: itself and the empty set. Its power set consists of these same sets.
The cardinality of a singleton set is one, meaning if A is a singleton, then |A|=1
The union of a singleton set with another set will simply result in the same singleton set or the set with the new added element, respectively.
The intersection of a singleton set with any other set will either consist of the same element or be empty, depending on whether they share that element or not.
The intersection between two singleton sets is the same set or the empty set, depending on whether the singleton sets are equal or not.
The complement of a singleton set in a universal set is the set containing all elements that are in the universal set but not in the singleton set.

Practice exercises

Exercise 1: If the set *A=\{2,a,b\}* is a singleton set, calculate *a+b.*

Solution:

Since A is a singleton set, all its elements are equal, that is, *2=a=b.* From here, we extract that *a=2* and *b=2,* therefore *a+b=2+2=4.*

Exercise 2: the sets *A=\{2m, 12, n+2\}* and *B=\{20, 5p, q\}* are singleton sets. Calculate the sum *m+n+p+q.*

Solution:

Since A and B are singleton sets, both have only one element, namely:

For A: *2m=12=n+2*

We can find *m* and *n* by solving the equations *2m=12* and *n+2=12,* hence *m=6,* *n=10.*

For B: *20=5p=q*

We can find *p* and *q* by solving the equations *20=5p* and *20=q.* From there, we extract that *p=4* and *q=20.*

Having the requested values, we can calculate *m+n+p+q=6+10+4+20=40*

Exercise 3: The set *A=\{a+b; b+c; a+c; 6\}* is a singleton set. Calculate *a+b+c.*

Solution:

Since A is a singleton set, all its elements are equal, that is: *a+b=b+c=a+c=6,* in the form of a system:

*\begin{cases} a+b=6 \\ b+c=6 \\ a+c=6 \end{cases}*

Solving the system, we find that *a=3,* *b=3,* and *c=3,* therefore *a+b+c=9.*

Exercise 4: Given the singleton sets: *A=\{x+7; 2x+5\}* and *B=\{y-3; 5y-15\},* what is the value of *x+y*?

Solution:

Since A is a singleton set, *x+7=2x+5* from which it follows that *x=2.* Since B is also a singleton set, it must be that *y-3=5y-15,* from which it follows that *y=3.* Having *x* and *y,* we can calculate *x+y=2+3=5.*

Daniel Machado

Advanced student of Mathematics at Facultad de Ciencias Exactas, Químicas y Naturales. Universidad Nacional de Misiones, Argentina.