Relation and types of relations
Relation and Types of Relations
In mathematics, a relation is a way of showing a connection or relationship between sets of information. Formally, a relation from a set $A$ to a set $B$ is a subset of the Cartesian product $A \times B$. The Cartesian product of two sets $A$ and $B$, denoted by $A \times B$, is the set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.
Definition of a Relation
A relation $R$ from a set $A$ to a set $B$ is defined as a set of ordered pairs, where the first element of each pair comes from $A$ and the second element comes from $B$. Mathematically, we can write this as:
$$ R \subseteq A \times B $$
For example, if we have two sets $A = {1, 2, 3}$ and $B = {4, 5}$, then a relation from $A$ to $B$ could be $R = {(1, 4), (2, 5), (3, 4)}$.
Types of Relations
There are several types of relations that are important in mathematics:
- Reflexive Relation
- Symmetric Relation
- Transitive Relation
- Equivalence Relation
- Partial Order Relation
Let's explore each of these types in more detail.
Reflexive Relation
A relation $R$ on a set $A$ is called reflexive if every element is related to itself. Formally, $R$ is reflexive if:
$$ \forall a \in A, (a, a) \in R $$
Symmetric Relation
A relation $R$ on a set $A$ is called symmetric if for every pair in the relation, the reverse of the pair is also in the relation. Formally, $R$ is symmetric if:
$$ \forall a, b \in A, \text{ if } (a, b) \in R \text{ then } (b, a) \in R $$
Transitive Relation
A relation $R$ on a set $A$ is called transitive if whenever a pair is in the relation and the second element of that pair is related to a third element, then the first element is also related to the third element. Formally, $R$ is transitive if:
$$ \forall a, b, c \in A, \text{ if } (a, b) \in R \text{ and } (b, c) \in R \text{ then } (a, c) \in R $$
Equivalence Relation
A relation $R$ on a set $A$ is called an equivalence relation if it is reflexive, symmetric, and transitive. Equivalence relations partition the set into equivalence classes.
Partial Order Relation
A relation $R$ on a set $A$ is called a partial order relation if it is reflexive, transitive, and antisymmetric (where antisymmetric means that if $(a, b) \in R$ and $(b, a) \in R$, then $a = b$).
Differences and Important Points
The following table summarizes the differences between the types of relations:
| Property/Type | Reflexive | Symmetric | Transitive | Equivalence | Partial Order |
|---|---|---|---|---|---|
| Reflexive | Yes | No | No | Yes | Yes |
| Symmetric | No | Yes | No | Yes | No |
| Transitive | No | No | Yes | Yes | Yes |
| Antisymmetric | No | No | No | No | Yes |
Examples
Here are some examples to illustrate the types of relations:
Reflexive Relation Example
Let $A = {1, 2, 3}$ and $R = {(1, 1), (2, 2), (3, 3)}$. This relation is reflexive because every element is related to itself.
Symmetric Relation Example
Let $A = {a, b}$ and $R = {(a, b), (b, a)}$. This relation is symmetric because for every $(a, b)$, there is a corresponding $(b, a)$.
Transitive Relation Example
Let $A = {1, 2, 3}$ and $R = {(1, 2), (2, 3), (1, 3)}$. This relation is transitive because $(1, 2)$ and $(2, 3)$ imply $(1, 3)$.
Equivalence Relation Example
Let $A = {1, 2, 3, 4}$ and $R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (3, 4), (4, 3)}$. This relation is an equivalence relation because it is reflexive, symmetric, and transitive.
Partial Order Relation Example
Let $A = {1, 2, 3}$ and $R = {(1, 1), (2, 2), (3, 3), (1, 2), (1, 3)}$. This relation is a partial order because it is reflexive, transitive, and antisymmetric.
Understanding these types of relations is crucial for various fields in mathematics, including set theory, algebra, and discrete mathematics. They form the foundation for more complex structures and concepts such as functions, groups, and lattices.