Relations and Functions

Introduction 

“All functions are relations, but all relations are not funtions.”- let us try to understand this statement by studying what are functions and relations. The relation and function both are the important topics of algebra as the function was introduced in the 17th century in 1637 by the modern philosopher and the mathematician Rene Descartes and discussed with many mathematicians then term it as the name “function” then the notations are introduced y=x so that we will denote the function f(x)= y and for the derivative denotation dy/dx. in function, there are two parts domain and codomain or we can say the range is present, as when we talk about relation it is a bunch or a collection of ordered pairs.

About relation and function

Relation and function 

A relation is a set of ordered pair numbers or we can say that a relation is a bunch or a collection of ordered pairs and when the first member repetition is associated with the repetition of the second member of the element then the relationship becomes the function as it contains the properties of the function at this time.

A function is a special type of relationship as we can say that a relation f from a set X to the other set Y is called a function as if every element of Set X has one and only one image of the other set Y and no distinct elements of the other set Y and then they will have a same mapped first element. Where we also got to know that X and Y both are non-empty sets. Where the X is a domain, and the whole set Y is a codomain. The function is represented as f:X->Y which is also written as f(x) = y, where (x,y) belongs to f and x belongs to X and y belongs to Y, which means the elements belong to their corresponding sets in function there are two parts domain and codomain are present domain is a set of all the inputs or the first value of the function which is generally introduced as “x”. Then the codomain is the collection of the outputs or the second values which it is denoted by “y”. 

Types of relation

A relation is a set of ordered pair numbers, or we can say that a relation is a bunch or a collection of ordered pairs therefore, it is divided into some types that are 

1. Empty relations: when the set contains no element or we can say when there is no element on the set or any element to map that relation is called empty and this is also called void relation.
2. Universal relation: when the set has a relation with every element or we can say have a full relationship with the set then this relation is called universal relation. Which is written as R= A*A, as every element of set A is inset B the this is a full relation of universal relation.
3. Identity relation: a relation is an identity relation when every element of a particular set is related to itself. For example, when we throw the dice, the total outcomes will be 36 that will be (1,5) (4.3) (2,2) (1,6) (3,3) therefore from this the outcomes like (1,1) (2,2) (3,3) this kind of outcomes are identity relations.
4. Inverse relation: the relation is called inverse relation. When the relation from is from set A to set B, then this is inverse relation R -1.
5. Reflexive relation: the relation is called reflexive if every element of Set A related to itself or we can sit maps itself.
6. Symmetric relation: in this set A (a,b) belongs to R and (b, a) also belongs to R then the relation is called symmetric relation.
7. Transitive relation: when (a,b) belongs to R and (b,c) also belongs to R, then if (a,c) should also belong to R if it happens, then the relation is transitive. 

Types of function

A function is a special type of relationship, as we can say that a relation f from a set X to the other set Y is called a function. This is also divided into some types here are they:

1. One to one function: this is also called injective function and the function is called injective when for each value of the domain there is a distinct element in the codomain or we can say each element of P has a distinct element in Q.
2. Onto function: this function is called surjective function and the function is called surjective when every element of set Q there is a preimage of set P.
3. One-one and onto function: this is also called bijective function and the function is called bijective when it follows both the conditions or the set is both injective and surjective then the function is bijective.

Conclusion 

From the above content, we learn that both relation and function are important topics of algebra. While sometimes people become confused by these two terms as they are completely different. We get to know that a relationship is a set of ordered pair numbers or we can say that a relation is a bunch or a collection of ordered pairs and a function is a special type of relationship as we can say that a relation f from a set X to the other set Y is called a function these both are divided into some types too.