Relations and Functions

Relations and Functions in real life give us the link between any two entities. In our daily life, we come across many patterns and links that characterize relations such as a relation of a father and a son, brother and sister, etc. In mathematics also, we come across many relations between numbers such as a number x is less than y, line l is parallel to line m, etc. Relations and functions map elements of one set (domain) to the elements of another set (codomain).

Functions are nothing but special types of relations that define the precise correspondence between one quantity with the other. In this article, we will study how to link pairs of elements from two sets and then define a relation between them, different types of relations and functions, and the difference between relations and functions.

Relations and functions define a mapping between two sets (Inputs and Outputs) such that they have ordered pairs of the form (Input, Output). Relations and functions are very important concepts in algebra. They are used widely in mathematics as well as in real life. Let us define each of these terms of relations and functions to understand their meaning.

Relations and Functions Definition

Relations and functions individually are defined as:

• Relations - A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
• Functions - A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.

Please note that all functions are relations but all relations are not functions.

Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphically, roster form, and tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2) = 4, f(3) = 9. Now, let us represent this function in different forms.

• Set-builder form - {(x, y): f(x) = y², x ∈ A, y ∈ B}
• Roster form - {(1, 1), (2, 4), (3, 9)}
• Arrow Representation -
  
• Table Representation -

Now that we have understood the meaning of relations and functions, let us understand the meanings of a few terms related to relations and functions that will help to understand the concept in a better way:

• Cartesian Product - Given two non-empty sets P and Q, the cartesian product P × Q is the set of all ordered pairs of elements from P and Q, that is, P × Q = {(p, q) : p ∈ P, q ∈ Q}
• Domain - The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the domain of the relation R. It is called the set of inputs or pre-images.
• Range - The set of all second elements of the ordered pairs in a relation R from a set A to a set B is called the range of the relation R. It is called the set of outputs or images.
• Codomain - The whole set B in a relation R from a set A to a set B is called the codomain of the relation R. Range ⊆ Codomain

There are different types of relations and functions that have specific properties which make them different and unique. Let us go through the list of types of relations and functions given below:

Types of Relations

Given below is a list of different types of relations:

• Empty Relation - A relation is an empty relation if it has no elements, that is, no element of set A is mapped or linked to any element of A. It is denoted by R = ∅.
• Universal Relation - A relation R in a set A is a universal relation if each element of A is related to every element of A, i.e., R = A × A. It is called the full relation.
• Identity Relation - A relation R on A is said to be an identity relation if each element of A is related to itself, that is, R = {(a, a) : for all a ∈ A}
• Inverse Relation - Define R to be a relation from set P to set Q i.e., R ∈ P × Q. The relation R^-1 is said to be an Inverse relation if R^-1 from set Q to P is denoted by R^-1 = {(q, p): (p, q) ∈ R}.
• Reflexive Relation - A binary relation R defined on a set A is said to be reflexive if, for every element a ∈ A, we have aRa, that is, (a, a) ∈ R.
• Symmetric Relation - A binary relation R defined on a set A is said to be symmetric if and only if, for elements a, b ∈ A, we have aRb, that is, (a, b) ∈ R, then we must have bRa, that is, (b, a) ∈ R.
• Transitive Relation - A relation R is transitive if and only if (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R for a, b, c ∈ A
• Equivalence Relation - A relation R defined on a set A is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive.

Types of Functions

Given below is a list of different types of functions:

• One-to-One Function - A function f: A → B is said to be one-to-one if each element of A is mapped to a distinct element of B. It is also known as Injective Function.
• Onto Function - A function f: A → B is said to be onto, if every element of B is the image of some element of A under f, i.e, for every b ∈ B, there exists an element a in A such that f(a) = b. A function is onto if and only if the range of the function = B.
• Many to One Function - A many to one function is defined by the function f: A → B, such that more than one element of the set A are connected to the same element in the set B.
• Bijective Function - A function that is both one-to-one and onto function is called a bijective function.
• Constant Function - The constant function is of the form f(x) = K, where K is a real number. For the different values of the domain(x value), the same range value of K is obtained for a constant function.
• Identity Function - An identity function is a function where each element in a set B gives the image of itself as the same element i.e., g (b) = b ∀ b ∈ B. Thus, it is of the form g(x) = x.

Important Notes on Relations and Functions

• Relations and functions define a mapping between two sets (Inputs and Outputs) such that they have ordered pairs of the form (Input, Output).
• Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphically, roster form, and tabular form.
• All functions are relations but all relations are not functions.

Topics Related to Relations and Functions

• Antisymmetric Relations
• Relations and Functions Worksheets
• Types of Functions
• Domain and Range of a Function

What are Relations and Functions in Math?

Relations and functions individually are defined as:

• Relations - A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
• Functions - A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.

What is the Difference between Relations and Functions?

The difference between relations and functions is that relations define any relationship between inputs and outputs whereas a function defines a relation such that each input has only one output. All functions are relations but all relations are not functions.

What is Range in Relations and Functions?

The set of all outputs obtained from a relation or a function is called range.

How Do you Represent Relations and Functions?

Relations and functions can be represented in different forms such as arrow diagram, algebraic form, set-builder form, graphically, roster form, and tabular form.

What are the Types of Relations and Functions?

There are different types of relations and functions such as empty relation, universal relation, reflexive relation, symmetric relation, transitive relation, equivalence relation, constant function, polynomial function, identity function, on-to-one function, onto function, bijective function, etc.

What are Ordered Pairs in Relations and Functions?

Relations and functions have ordered pairs of the form (Input, Output). The input belongs to the domain and output belongs to the range of the relation/function.

How to Determine if a Relation is a Function?

if in a relation, each element in the domain is mapped to a unique element in the codomain, then it is said to be a function.

How to Identify Relations and Functions?

A binary operation defining a link between a set of elements with another set of elements is a relation. But if each element of the first set is mapped to one and only one element of the second set, then it is a function.