Type of functions in the sets and relations

In Day-to-day life, relations and functions provide a link between two entities. We come across various patterns and links that describe the connection of sets and functions in our daily lives, such as the relation between a parent and a son, brother, and sister, and so on. We come across various number relations in mathematics also, such as x is less than y, line l is parallel to line m, and so on. Relation and functions connect elements of one set (domain) to those of another (codomain). 

Functions are specialized sorts of relations that define the exact correspondence between two quantities. In this post, we’ll look at how to connect pairs of elements from two sets and then establish a relation between them, as well as the various types of relation and function and the distinction between them. 

Definition of Relation and Function 

Let us define the Relation and function individually:

• Relation: A subset of the cartesian product A B is a relation R from a non-empty set B. The subset is calculated by characterizing the relation between the first and second elements of the ordered pairs in A.

• Function: If every element of set A has one and only one image in set B, the relation f from set A to set B is called a function. To put it another way, there are no two separate pieces of B that share the same pre-image.

We can say that a function is a special type of relation.

TYPES OF RELATION: – 

1. EMPTY RELATION: –

There are no items in these types of relations, hence it’s an empty set. It means that an element from one set A to the other set B has no association or mapping. And this empty relation is denoted by R=ϕ(phi).

2. UNIVERSAL RELATION

 It’s the opposite of an empty relationship. From and to the same set A, this relation is defined. Every element in A has a relation with every other element in A. In a nutshell, it can be expressed as R=A*B

3. IDENTITY RELATION

The identity relation is one in which each element exclusively maps to itself. It’s the relation in which each element of A is simply related to itself. 

Consider the relation R, which is defined on the set A. If R denotes an identity relation, then

R = (a, a) / for all a belongs to A

4. REFLEXIVE RELATION

  If every element of the first set A is related to itself, the connection is reflexive. R is reflexive in mathematics if (x, x)R is true for all x €A.

5. SYMMETRIC RELATION

A relation R is said to be a symmetric relation, if for every x and y If (x,y)∈R then it implies (y,x)∈R, for all x∈A and y∈B.

6. TRANSITIVE RELATION

A relation R is said to be a transitive relation, if for every x , y, z  (x,y)∈R and (y,z)∈R, implies (x,z)∈R, where x,y,z∈A. 

An example of the transitive relations “Y is less than X”. 

7. INVERSE RELATION

 The relation S said to be an inverse relation of a given relation R, if (y,x)∈S for all (x,y)∈R 

8. EQUIVALENCE RELATION

 If a relation is reflexive, symmetric, and transitive, it is said to be an equivalence relation. 

For example, on a set A of all lines in a plane, the relation “is parallel to” is an equivalence relation. Because of the following reasons: Every line is parallel to itself, demonstrating reflexivity. If line l is parallel to line m, then m is parallel to l as well. If l is parallel to m and m is parallel to n, l must be parallel to n as well. 

CONCLUSION:-

The relation is a subset of the cartesian product of A B is a relation R from a non-empty given set B. The subset is calculated by characterizing the relation between the first and second elements of the ordered pairs in A and B . Set theory and related operations rely heavily on the concept of relations. As a result, they’re crucial in other ideas like functional analysis. The applications are numerous, and they lay the groundwork for many other areas of set theory.