Relation And Function

In real life, the terms “relations” and “function” indicate the connection between any two entities. In our daily lives, we come across various patterns and links that characterise relationships such as the relationship between a father and a son, or the relationship between a brother and a sister, among others. In mathematics, we come across many relationships between numbers, such as the fact that a number x is less than a number y, the fact that line l is parallel to line m, and so on. The terms relation and function are used to map items of one set (domain) to elements of another set (domain) (codomain).

When it comes down to it, functions are nothing more than particular forms of relationships that determine the precise correspondence between two quantities. Throughout this article, we will look at how to link pairs of elements from two sets and then establish a relation between them, the many forms of relation and function, and the distinction between relations and functions.

Relation and Function Definition

Relations – A relation R derived from a non-empty set B is a subset of the cartesian product A × B derived from the relation R. By explaining a relationship between the first element and the second element of the ordered pairs in A × B, we can obtain the subset.

 Functions – When a relation f from a set A to a set B is considered to be a function, it means that each and every element of set A has exactly one and only one image in set B. In other words, there are no two unique pieces of B that have the same pre-image as another.

Types of Relations

The following is a list of the various sorts of relationships that can exist:

• The term “empty relation” refers to a relation that is devoid of any elements; that is, a relation in which no element of set A is mapped or linked to any element of A. It is symbolised by the symbol R = ∅.
• A relation R is universal if each element of A is related to every other element of A, i.e., R = A A. It is referred to as the Universal relation.
• It is considered to be an identity relation if each element of A is connected to itself, i.e. R = {(a, a): for all a ∈ A}.
• Inverse Relation -Define R as a relation from set P to set Q, i.e., R ∈ P × Q.

• In a reflexive relation R defined on a set A, aRa is said to be present for every element a, that is (a, a) ∈ R.
•  In order for a binary relation R defined on a set A to be described as symmetric, it must satisfy the following two conditions: first, we must have aRb, i.e., (a, b)  ∈ R, and second, we must also have bRa, i.e., (b, a) ∈  R for elements a and b ∈ A.
• A relation R is transitive if and only if (a, b)  ∈ R and (b, c)  ∈ R (a, c) ∈  R for all a, b, and c ∈   A
• An equivalence relation is defined on a set A when and only when it is reflexive, symmetric, and transitive, and when and only when it is defined on a set A .

Types of Functions

The following is a list of the various sorts of functions available:

• As the name suggests, an A->B function that maps each element of A to a single component of B is known as a one-to-one function. It is also referred to as Injective Function.
• When a function A:  → B is said to be onto, then every element of B is the image of some element of A under f, that is, for every b: B, there exists an element of A such that f(a) = B. Onto Functions are used in the analysis of functions. An onto function is defined as follows: If the range of the function equals B, then the function is onto.
• Many to one functions are defined by the f: A →  B function, which connects several elements from the first group (A) to the second group (B).
• Bijective Function – A bijective function is a function that is both a one-to-one and an onto function in the same context.
• Constant Function – The constant function has the form f(x) = K, where K is a real number.  For a constant function, the same range value of K is achieved for a variety of distinct values of the domain(x value).
• Identity Function – An identity function is a function in which each element in a set B gives the image of itself as the same element, i.e., g (b) =  b ∀ b ∈ B.  As a result, it has the form g(x) = x.

CONCLUSION

A relation between two sets is a collection of ordered pairs containing one object from each set that represents the relationship between the sets. It is said that the objects are connected if the ordered pair (x,y) exists in the relation between the objects if the items are both from the first set and both from the second set. It is possible to have more than one relationship at the same time. The vertical line test can be used to determine whether a relationship is a function by charting the numbers on a graph and applying the results. Any relation that does not have more than one point of intersection with a vertical line flowing across the graph is called a function.