An Introduction to Relation and Function
Relations and functions are both closely related to each other. One needs to have a clear knowledge to understand the concept of relations and functions to be able to differentiate them. In this article, we are going to distinguish between relations and functions.
Two or more sets can be related to each other by any means known as Relation.
Let us consider an example of two sets A and set B having m elements and n elements, respectively., we can easily have a relation with any ordered pair which shows a relation between the two sets A and B.
A function can have the same range mapped as that of in relation, such that a set of inputs is related with exactly one output.
Let us consider an example. Set A and Set B are related in a manner that all the elements of Set A are related to exactly one element of Set B or many elements of the given set A are related to one element of given Set B. Thus, this type of relation is known as a function.
We see that a given function cannot have one to many relation between the set A and set B.
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What is a Relation?
Relation in Mathematics can be defined as a connection between the elements of two or more sets, the sets must be non-empty.
A relation R is formed by a Cartesian product of subsets.
For example, let us say that we have two sets, then if there is a connection between the elements of two or more non-empty sets, the only relation is established between the elements.
Different Types of Relations in Mathematics
There are different types of relations in math which define the connection between the sets.
There are eight types of relations.
Empty Relation – We can write an empty relation as R = ∅.
Universal Relation – Universal relation can also be known as a full relation as every element of set A is related to every element in B. An empty and universal relation can also be named as a trivial relation. A universal relation R in a set, say A, is one in which each member of A is connected to every other element of A, i.e. R = A x A.
Identity Relation – A relation is called an identity relation if every element of set A is related to itself only. We have 36 potential results if we roll two dice: (1, 1), (1, 2), ..., (6, 6). It is an identity connection if we describe it as R: (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6).
Inverse Relation – Suppose we have a relation R from set A to set B, R∈ A×B. Then, the inverse relation of R can be written as R-1 = {(b, a) : (a, b) ∈ R}.
Reflexive Relation – If every element of set A maps for itself, then set A is known as a reflexive relation. For every a ∈ A, (a, a) ∈ R. If the set A={1,2,3}, then the relation {(1,1),(2,2),(3,3)} is a reflexive relation.
Symmetric Relation – A relation R on a set A is known as a symmetric relation if (a, b) ∈R then (b, a) ∈R, such that for all a and b ∈ A.
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.
Equivalence Relation – A relation is said to be an equivalence relation if (if and only if) it is transitive, symmetric, and reflexive.
NOTE: All functions are relations, but not vice versa.
What is a Function?
A function is a relation that says that there should be only one output for each input.
In simpler words, we can say that it is a special kind of relation or a set of ordered pairs that follow a rule that every value of x should be associated with only one value of y. This is known as a function.
Types of Functions-
In terms of relations, the types of functions can be defined as:
1. One-to-one Function:
A function f: A → B is said to be One to One if for each value of A there is a distinct value of B.
The one-to-one function is also known as the iInjective function.
Example:
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In the above example, we see that a single element can map only one element.
2. Many-to-one Function:
A many-to-one function is one that maps two or more elements of A to the same element of set B.
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Here, in this example, we see that a single element can map one or more elements.
3. Onto Function:
A function for which every element of set B there is pre-image in set A is known as onto function.
The onto function is also known as subjective function.
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In this example, we see that each and every element of B is used and this denotes the onto function.
4. One-one and Onto Function:
The function f matches with each element of A with a discrete element of B and every element of B has a pre-image in A.
The one-one and onto function is also known as bijective function.
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In this example that denotes the bijective function, each element maps only one element and all of the elements of Y are mapped, thus satisfying our condition.
Some Special Functions in Algebra-
1. Constant Function – A fixed value is referred to as a constant value. A constant in Algebra is a number, although it may also be a letter such as a, b, or c for a given integer. x+2=10, for example, where 2 and 10 are constants. We can write a constant function as f(x) = c.
2. Identity Function – An identity function, also known as an identity relation, identity map, or identity transformation, is a function that returns the same result as its input every time. That is, if f is identity, then f(X) = X holds true for every X. We can write a constant function as f(x) =x.
3. Absolute Value Function – A function having an algebraic expression enclosed in absolute value symbols is known as an absolute value function. We can write an absolute value function as f(x) =|x|.
4. Inverse Functions – A function that "undoes" another function is called an inverse. That example, if f(x) yields y, then putting y into the inverse of f generates the output x. Invertible functions are those that have an inverse, and the inverse is denoted by f-1. We can write an inverse function as f-1 (x).
5. Linear Functions – An equation with a variable, its derivative, plus a few additional functions is known as a linear differential equation. A linear differential equation's typical form is dy/dx + Py = Q, which includes the variable y and its derivatives. We can write a linear function as f(x ) = mx+c.
Difference between Relations and Functions
The above table shows the difference between relations and functions in Mathematics.
Relations and Functions Notes:
There are three ways to represent a relation in mathematics.
1. Roster Form – Roster form is basically a representation of a set which lists down all of the elements present in the set and are separated by commas and enclosed within braces.
2. Set-Builder Form – A shorthand method is used to write sets and is often used for sets with an infinite number of elements. It is used with different types of numbers, such as integers, real numbers, and so on. The set-builder form is also used to express sets with an interval or an equation.
3. By Arrow Diagram – In the by arrow diagram method, the relation between sets is denoted by drawing arrows from the first components to the second components of all the pairs which belong to the relation.
Questions to be solved:
Question 1) What is the sum of two functions given below:
f(x) = 2x+3 , g(x) = 4x+4
Answer),
(f+g)(x) = 2x+3+4x+4
=6x+7
FAQs on Difference Between Relations and Functions for JEE Main 2024
1. What is the difference between functions and relations in maths and Is every relation a function?
A relation is defined as a set of inputs and outputs, and a function is defined as a relation that has one output for each input.
For every finite sequence of objects which are known as the arguments, a function associates a unique value. In fact, every function is basically a relation. However, not every relation can be known as a function.
2. Are functions and relations different?
Yes, functions and relations are different because functions are a special form of relation a set possesses while the absence of it gives it a name as relation.