A set is a collection of distinct objects, considered as an object in its own right.
A singleton is a set with exactly one element, and an empty set is a set with no elements, denoted by .
It is defined as
If and are sets, then is a subset of , denoted by , if every element of is also an element of .
Two sets and are equal, denoted by , if they have the same elements, i.e., and .
Set operations include:
In a specific theory, we often have a universal set that contains all the objects under consideration.
The power set of a set , denoted by , is the set of all subsets of .
For example, if , then .
By Cantor's theorem, the power set of a set has a strictly greater cardinality than itself.
For example, the set of natural numbers is countably infinite, while its power set is uncountably infinite.
The Cartesian product of two sets and , denoted by , is the set of all ordered pairs where and :
Definition 1.1.1 (Relation) A relation from a set to a set is a subset of the Cartesian product .
If , we say that is related to by , denoted by .
An equivalence relation on a set is a relation on that satisfies the following properties for all :
Reflexivity: .
Symmetry: If , then .
Transitivity: If and , then .
The equivalence class of an element under the equivalence relation is the set of all elements in that are related to :
Proposition 1.1.2 If is an equivalence relation on a set , then , either or .
Proof. Let be an element in . Obviously, either or .
Let's consider the two cases.
First, if , then by definition of equivalence class, .
Since , we also have , and by symmetry .
By transitivity, we have .
Then, for any , we have . By transitivity, , so .
Thus .
By symmetry of the argument, we also have .
Therefore, .
Second, we have . For any , we have .
By transitivity, if , then and thus since , we have , which contradicts the assumption that .
Therefore, for all , and thus .
Thus, we have shown that either or .
There are some examples of relations:
The equality of real numbers is an equivalence relation.
Comparison of real numbers (i.e., , , , ) is not an equivalence relation.
Congruence modulo on the set of integers is an equivalence relation.
Similarity of triangles is an equivalence relation.
In electromagnetism, two vector potentials and are related by a gauge transformation if there exists a scalar function such that . This relation is an equivalence relation.
Definition 1.1.4 (Partition) A partition of a set is a collection of non-empty subsets of such that