Mathematical Proof/Relations

Introduction
Intuitively, relation associates pairs of objects based on some rules and properties. That is, relation suggests some kinds of relationship or connection between two objects. Take marriage as an example. In the marriage registry, there is a record in which the names of husbands are associated with the names of their corresponding wives, to keep track of the marriages. The entries in the record can be interpreted as $$(a,b)$$, where $$a$$ is the husband, while $$b$$ is the wife.

Expressing it mathematically, let $$M$$ and $$W$$ be the set of all men and women respectively. Then, the $$M\times W=\{(a,b):a\in M\text{ and }b\in W\}$$ consists of  pairs of people (first and second coordinate is a man and woman respectively). After that, we know that the record in the marriage registry, $$R$$, is a subset of $$M\times W$$. If a man and a woman form an ordered pair in $$R$$, say $$(m,w)$$, then it means they are married. Then, it is natural to say that $$m$$ is $$w$$. In other words, if we find that $$(m,w)\in R$$, then it means that they are related. Also, knowing what $$R$$ exactly is (we have the record from the marriage registry) is the same as knowing all the husband and wife relationship.

Thus, it is natural to define the set $$R$$ as a. Let us formally define below.

Reflexive, symmetric and transitive relations
After introducing the terminologies related to relations, we will study three properties for a relation defined a set.

Equivalence relations and equivalence classes
After studying the three properties that a relation on a set can possess, let us focus on those relations that possess properties.

Suppose $$R$$ is an equivalence relation on a set $$A$$. Intuitively, for elements that are related by $$R$$, they are quite "closely related". Thus, when we consider the set consisting elements that are related to a given element of set A, the elements inside the set are "closely related", so the set, in some sense, forms a "group" of elements that are "relatives". It then appears that we can classify the elements of set $$A$$ into different such "groups", according to an equivalence relation. As we will see, this is roughly the case. Hence, such "group" is quite important. Now, let us formally define what the "group" is:

{{colored example| Let $$A=\{1,2,3,4\}$$, and let a relation defined on $$A$$ be $$R=\{(1,1),(2,2),(3,3),(4,4),(1,2),(1,3),(2,1),(3,1),(2,3),(3,2)\}.$$ This relation $$R$$ can be shown to be an equivalence relation. The equivalence classes are given by $$[1]=\{1,2,3\},[2]=\{1,2,3\},[3]=\{1,2,3\},[4]=\{4\}.$$ Since $$[1]=[2]=[3]$$, there are only two {{colored em|distinct}} equivalence classes. Graphically, the situation looks like: ^       ^  |        | [1]=[2]    [4] =[3] {{colored exercise| Construct an equivalence relation $$R$$ on $$A$$ such that the equivalence classes are given by $$[1]=\{1,2\},[2]=\{1,2\},[3]=\{3,4\},[4]=\{3,4\}.$$ {{hide|Solution| $$R=\{(1,1),(2,2),(3,3),(4,4),(1,2),(2,1),(3,4),(4,3)$$. Graphically, the situation looks like: ^       ^  |        | [1]=[2]  [3]=[4] }} }} }}
 * 2  3  /  |
 * 1   /  4 |
 * 2  3  /  |
 * 1   /  4 |
 * 1   /  4 |
 * 2 | 3    |
 * 1 \    4 |
 * 2 | 3    |
 * 1 \    4 |
 * 1 \    4 |

Properties of equivalence classes
In this section, we will discuss some properties of equivalence classes. In particular, we will address these two questions: The answer to question 1 is given by the following theorem.
 * 1) When are two equivalence classes equal?
 * 2) Can two different equivalence classes contain a common element?

Now, let us consider the question 2. The answer to question 2 is, indeed, "No". The following corollary justifies this answer:

Now we have reached a key point in studying equivalence relations (and it is probably the major reason for studying equivalence relations at all): using equivalence relation on a set to construct a partition of that set, and vice versa. Before discussing it, let us define of a set:

We can observe from the previous examples that equivalence relation of a set can be used to give a partition of that set. The following theorem suggests that, in general, an equivalence relation on a set $$A$$ can be used to give a partition of that set.

The following theorem suggests the converse of the above theorem is also true. To be more precise, we can use a partition of a set to construct an equivalence relation on that set. Before introducing the theorem, let us make some intuitive guesses on how to construct the equivalence relation in this way. First, from the previous theorem, roughly speaking, using an equivalence relation on a set, we can create several "groups" of elements in different classes, in which the elements are "relatives".

Now, given a partition of a set, it means we have several "groups" of elements. Such "grouping" intuitively indicates the elements inside the group are "relatives" in some sense. So, intuitively, a relation that relates the "relatives" seems to make the relation quite "close", and hence an equivalence relation.

The following theorem formalizes this intuition: