All About Relations and Functions Complete Relation

Relations and functions are extremely important topics of mathematics that have their use in almost every field of science. In science, sets, relations and functions go hand in hand. Though they are closely related, while all functions are essentially some sort of relation the converse is not true. 

A relation in mathematics is defined as a set of ordered pairs where the relationship between the values of the ordered pair is defined as a relation. To put it simply a relation defines how the x and the y value are related. This could be based on some equation where y is a dependent variable and x is the independent variable or it can be that y is expressed as a function of x. Therefore, a relation is nothing but how two different sets of values/ numbers are related to each other. Subsets of two Cartesian products are also a relation. 

Different Types of Relations

In mathematics we come across various types of relations. The 7 different types of relations that we encounter include Empty relations, universal relations, identity relations, inverse relations, reflexive relations, symmetric relations and transitive relations. 

• Empty Relations: An empty relation is one where none of the elements of the first set is related to any of the elements of the other set. If we consider two sets, set A and set B then when no element of set A can be mapped to any of the values of set B, the relation is said to be an empty relation. An empty relation is also termed a null or a void relation. The mathematical representation of an empty relation is $\mathfrak{R}=\boldsymbol{\Phi}$

• Universal Relation: An universal relation is that type of a relation between two sets where all the elements of the first set are present in the second set. For example, if we consider two sets, A and B then all the elements of set A when present in set B, set A and set B are said to have a universal relation. A universal relation is mathematically expressed as $\mathfrak{R}=A \times B$.

• Identity Relation: An identity relation is one where every element of a set is related to or can be mapped to itself. The ordered pairs essentially have the same value. Let us take the example of two coins. Flipping a coin we can get four outcomes (H, H), (H, T),(T, H) and (T, T). If we consider the ordered pairs {(H,H) ,(T,T)} then it will be an identity relation. An identity relation is expressed as , $I=\{(A, A), \in A\}$

• Inverse Relation: In the case of an inverse relation, if we consider the two sets set A and set B, then the domain of one set will be the range of the other and vice versa. In the case of an inverse relation, the elements of set A can be Mapped to the elements of set B and similarly the elements of set B can be related to those of set A. The mathematical representation of an inverse relation is as follows $R^{-1}=\{(b, a):(a, b) \in R\}$ 

• Reflexive Relation: For any set of numbers, if the elements are mapped back to themselves, the relation is termed a reflexive relation. A reflexive relation might sound similar to an identity relation. However, the key point of difference between the two is that for an identity relation the elements are only related to themselves but for a reflexive relation that is not so. For any set X, if $x\in X$, THEN $(a, a) \in \mathfrak{R}$

• Symmetric Relations: Symmetric relations are those where even if the order of the ordered pairs changes, the relation remains the same. This means that if $(x, y) \in \mathfrak{R}$ then $(y, x) \in \mathfrak{R}$

• Transitive Relation: This type of relation shows that for any set X, if 

$\left(x_{1}, x_{2}\right) \in \mathfrak{R}$ and $\left(x_{2}, x_{3}\right) \in \mathfrak{R}$ then $\left(x_{1}, x_{3}\right) \in \mathfrak{R}$. 

• Equivalence Relation: When one particular relation is reflexive, symmetric as well as transitive at the same time, the relation is termed an equivalence relation. 

• Complete Relation: For two sets X and Y, a complete relation will be the entire set of $X\times Y$. A complete relation is always reflexive in nature along with being transitive and symmetric. The domain of the source set X is given by $\{x: \text { there is a y with } x \mathfrak{R} y\}$ 

Difference Between Relations and Functions

As mentioned before,  all functions are relations because they establish some kind of a relationship between the independent variable and the dependent variable. However, the converse is not true. This means that all relations do not necessarily have to satisfy a function to get mapped. The main points of difference between a relation and a function are as follows:

Conclusion

Relations are the relationships established between an element of one set and another element of the second state. A relation can also be defined as the subset of the Cartesian product of two individual sets. A function is also a relation where every individual input has a discrete output. There are different types of relations in mathematics that one can come across. Empty relation, universal relation, symmetric relation, transitive relations and reflexive relations are some of the most important types. In relations and functions, complete relation is one which exhibits the properties of reflexivity, transitivity and symmetry.