Set Theory Notation
Set theory was developed by mathematicians to be able to talk about collections of objects. It has
turned out to be an invaluable tool for defining some of the more complicated mathematical structures. As
such, we will be making use of some very basic set theory, mostly as a convenient notation for some of our
constructions. If you have seen this material before, feel free to ignore this handout. If you have further
questions about this material, I would be happy to provide further explanation or examples.
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Sets
A set is just a collection of mathematical objects. These objects can be numbers, vectors, spaces, or even
other sets.
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Elements
The objects in these collections are called elements. Given a set S and an object x, we use the notation
x ∈ S to denote that x is an element of S, and x 6∈ S to denote that x is not an element of S. When x ∈ S,
we say that x is an element of S or that S contains x.
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Defining Sets
We have three basic ways of writing down a set. We can list all of the elements, write down some defining
property for its elements, or write its elements as the values taken by some expression.
3.1
Lists of Elements
The most basic way to define a set is as a list of its elements placed between curly braces. So {1, 2, 3} is
the set whose only elements are 1, 2, and 3. The set {1, 2, 3, . . . , 100} has elements consisting of all integers
from 1 to 100, and {1, 2, 3, . . .} has elements consisting of all of the positive integers. One particular set of
interest, is the empty set. This is the set with no elements and it is denoted {} or ∅.
3.2
Elements Satisfying a Condition
To express more complicated sets, we can take the set of all elements of some other set T satisfying some
property P , written as S = {x ∈ T : P (x)}. This is the set of all elements x ∈ T so that the property P (x)
holds. The above set S would contain exactly the elements y so that y ∈ T and P (y) holds.
For example,
{x ∈ R : x > 1}
is the set of all real numbers bigger than 1 and
{(x, y) ∈ R2 : x = y}
is the set of all pairs of real numbers (x, y) so that x = y.
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3.3
Elements of a Given Form
Suppose that we want to express the set of all even numbers. That is the set of all integers n so that n is
equal to 2m for some other integer m. We could of course write this as
{n ∈ Z : there exists m ∈ Z so that n = 2m}.
On the other hand, this notation is somewhat cumbersome. It is often useful to have a way to write the set
of all objects that can be produced in some way. Thus, for the set of even integers we use the alternative
notation:
{2m : m ∈ Z}.
In general we use the notation
{f (x, y, z) : x ∈ S, y ∈ T, z ∈ R}
to denote the set of all things that can be written as f (x, y, z) for some elements x, y, z of the sets S, T, R,
respectively. So for example,
{n2 + m2 : n, m ∈ Z}
is the set of all numbers that can be written as the sum of the squares of two integers, and
{{n, x} : n ∈ Z, x ∈ R}
is the set of sets that contain as elements one integer and one real number.
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Equality and Subsets
Two sets are considered to be equal if they contain exactly the same elements. In other words two sets, S
and T , are equal if and only if the elements of one are exactly the same as the elements of the other. So
S equals T if for all x, x ∈ S if and only if x ∈ T . This definition has some important consequences. In
particular, it means that the sets {1, 2, 3}, {3, 1, 2}, and {1, 1, 2, 2, 2, 3} each contain only the elements 1,2,
and 3 and are thus all the same set.
If one cares about the order and multiplicity of elements, one will often consider tuples instead of sets.
A tuple is a sequence of elements for which both the order and the multiplicities matter. A tuple is written
as a list of elements between parentheses. For example (1, 2, 3), which is distinct from (3, 1, 2).
Another useful concept is that of subsets. We say that S is a subset of T (denoted S ⊂ T or S ⊆ T ) if
every element of S is also an element of T . So, for example, {1, 3} is a subset of {1, 2, 3}, but {1, 2, 5} is not
because 5 6∈ {1, 2, 3}.
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Operations
Another important way to create sets is to define them in terms of simpler sets. Here are a few simple
operations that can be used to do this.
Given two sets S and T the union, denoted S ∪T is the set of all elements contained in either set. Namely,
S ∪ T = {x : (x ∈ S) or (x ∈ T )}.
Given two sets S and T the intersection, denoted S ∩ T is the set of all elements contained in both sets.
Namely,
S ∩ T = {x : (x ∈ S) and (x ∈ T )}.
The difference of S and T , denoted S\T is the set of elements that are in S but not T . Namely
S\T = {x : (x ∈ S) and (x 6∈ T )}.
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Functions
One final piece of notation we’ll be making use of is that of functions. A function is a rule for assigning to
each element x in a set S, another element f (x) in some other set T . This f would be a function taking
elements of S to elements of T denoted f : S → T . In this case, S is called the domain of f and T is called
the codomain.
The function f is called surjective or onto if every element of the codomain can be written as some value
of f . Or in other words if for every y ∈ T there exists an x ∈ S so that f (x) = y.
The function f is called injective or one-to-one if distinct elements of the domain get mapped to distinct
elements of the codomain. In other words, f is injective when f (x) = f (y) only when x = y.
If f is both injective and surjective, it is bijective. This means that for each element of T that there is
one and only one element of S that f maps to it. In other words, f is bijective if for every y ∈ T there is a
unique x ∈ S so that f (x) = y.
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