Basic Set Theory

Set theory gives us the fundamental way to discuss many mathematical objects used in physics. We cover the basic concepts that we will need to discuss the mathematical structures in physics, such as groups, vector spaces, and manifolds.

Sets and Elements

Sets

A set is a collection of mathematical objects, that can be defined rigourously. Such objects in a set are called elements of the set.

Subsets

Take two sets and , as well as an element of , say . We denote that is an element of by , and that is not an element of by . If every element of is also an element of , we say that is a subset of , and denote it by . More formally,

If is a subset of , but is not equal to , we say that is a proper subset of , and denote it by .

Also, note that if and , then . This means that is a subset of itself.

Complement of a Set

Given a set and a subset , we can define the complement of in , denoted by , as the set of all elements in that are not in :

Power Set

Given a set , we can define the power set of , denoted by , as the set of all subsets of :

Note that the power set of includes both the empty set and the set itself, since both of them are subsets of .

Special Sets

There are a few special sets we will encounter in both mathematics and physics.

• The empty set is the set that has no elements.
• The set of natural numbers is the set of all positive integers, i.e. .
• The set of integers is the set of all integers, i.e. .
• The set of rational numbers is the set of all numbers that can be expressed as a fraction of two integers, i.e. .
• The set of real numbers is the set of all numbers that can be represented on the number line, including both rational and irrational numbers.
• The set of complex numbers is the set of all numbers that can be expressed in the form , where and are real numbers, and is the imaginary unit satisfying .

Union and Intersection of Sets

Given two sets and , we can define the union of and , denoted by , as the set of all elements that are in or in (or in both). Formally,

The intersection of and , denoted by , is the set of all elements that are in both and . Formally,

When the intersection of two sets is empty, we say that the two sets are disjoint, then union of two disjoint sets is denoted by .

(Cartesian/ Direct) Product of Sets

Given multiple sets, we can define the Cartesian/ direct product of those sets. Let be a set for , then the Cartesian product of those sets is defined as

We may also denote this as follows:

In these notations, the set is called the index set, where for each index in , we have a corresponding set .

Maps

Definition of Maps

Given two sets and , a map from to is a relation that assigns to each element of a unique element of . We denote such a map as follows:

Here, is called the domain of the map , and is called the range of . is called the image of . When is a set of numbers such as or , we may also call a function.

Images and Inverse Images

Given a map , we can define the image of a subset under as follows:

Similarly, we can define the inverse image of a subset under as follows:

Injective, Surjective, and Bijective Maps

Injective Map

A map is called injective (or one-to-one) if for ,

or equivalently,

Surjective Map

A map is called surjective (or onto) if

or equivalently,

Bijective Map

A map is called bijective if it is both injective and surjective. In that case, we can define the inverse map :

Equivalence Relation

Definition of Equivalence Relation

Given a set , an equivalence relation on is a relation that satisfies the following three properties:

1. Reflexivity: For all , .
2. Symmetry: For all , if , then .
3. Transitivity: For all , if and , then .

Equivalence Classes and Quotient Sets

Equivalence Class

Given an equivalence relation on a set , we can define the equivalence class of as follows:

We say that represents the equivalence class .

Quotient Set

The set of all equivalence classes of under the equivalence relation is called the quotient set of by , denoted by :