# Relations, Functions and their Differences

The terms “relations” and “functions” are interchangeable. To distinguish between relations and functions, one must have a thorough understanding of the concepts. We will see the difference between relation and function in this post.

## What is the definition of a relation?

The relation is the term used to describe how two or more sets may be linked together in any way. Consider the following scenario. We can easily connect with any ordered pair that demonstrates a relationship between the two sets A and B if they have m and n members, respectively.

## What is the definition of function?

A function’s range can be translated to that of in relation, resulting in a relation between a set of inputs and exactly one output. Consider the given scenario. Set A and Set B are linked so that all of Set A’s elements are related to exactly one element of Set B, or many elements of Set A is tied to one element of Set B.

As a result, this form of relationship is referred to as a function. We can see that a function can’t have a one-to-many relation between sets A and B.

## Functions and Relations Notes

In mathematics, there are three methods to describe a relation.

1. Roster Form – A roster form represents a set in which all of the components in the set are listed, separated by commas, and encased in braces.
2. Set-builder Form – A shorthand approach for writing sets, especially those with an unlimited number of components. It may be used with various numbers, including integers, real numbers, and so on. Sets containing an interval or an equation are also expressed using the set–builder form.
3. By Arrow Diagram – The by arrow diagram approach denotes the relation between sets by drawing arrows from the first to the second components of all the pairs that correspond to the connection.

## Different Types of Relations in Mathematics

In mathematics, several types of relations define the relations between sets.

There are eight different relations.

1. Empty relation: R = (Empty Relation) – We can write an empty Relation R = (Empty Relation).
1. Universal Relation: Universal Relation, also known as a Full Relation, is one in which every element of set A is connected to set B. A trivial relation is a relation that is both empty and universal.
1. Inverse Relation: Assume we have an R AB relation between sets A and B. The inverse relation of R may thus be expressed as R -1 = (b, a): (a, b) R.
1. Identity relation: If every member in set A is exclusively related to itself, it is termed an identity relation.
1. Reflexive Relation – A reflexive relation is one in which every element of set A maps for itself.
1. Symmetric Relation: A relation R on a set A is known as asymmetric relation if (a, b) ∈R then (b, a) ∈R, such that for all a and b ∈A.
1. Transitive Relation: A relation R in a set A is said to be transitive if (a, b) ∈R, (b, c) ∈R., then (a, c) ∈R such that for all a, b, c ∈A.
2. Equivalence Relation: A relation is said to be an equivalence relation if (if and only if) it is Transitive, Symmetric, and Reflexive.

## Types of Functions

The types of functions may be described as follows in terms of relations:

1. One-to-One Function: if there is a distinct value of B for each value of A, the function f: A B is said to be One to One. The Injective function is another name for the one-to-one function.

1. Many to One Function: A function that translates two or more items of set A to the same element of set B is known as a many to one function.

1. Onto Function: The Onto Function is a function for which every element of set B has a pre-image in set A. The subjective function is another name for the onto function.

1. One-one and Onto Function: Each element of A is matched with a discrete element of B by the function f, and every element of B has a pre-image in A.

## Relations vs Functions

 Relations Functions A relation is defined as a link between two or more groups of values. It may also be thought of as a subset of the Cartesian product. A function is defined as a relation in which each input has just one output. The letter “R” is used to represent a relation. A function is usually denoted by the letters “F” or “f.” Every relation, we may remark, is not a function. Every function in mathematics may be described as a relation

## Conclusion

In arithmetic, it’s not always simple to tell the difference between functions and relations. An ordered pair is a collection of inputs and outputs that, in essence, show a connection between two values.

A set of inputs and outputs is defined as a relation, and a function is defined as a relation with one output for each input. A function assigns a unique value to every finite sequence of objects known as the parameters. Every function is essentially a connection. However, not all relationships may be classified as functions.