Relations and Functions

Introduction to Relations and Functions

Let’s learn about “Relations and Functions” a topic which is a foundation or fundamental of algebra in math. Relations and functions – these are the two different words having different meaning mathematically. You might get confused about their difference. So, let’s study both these concepts with detailed explanation.
A kind of relations also exists in algebra which resembles the relations in our daily life. In daily life, relations are like a parent, siblings, friends, teachers, and students and many more. In math also, you see few relations like a line is perpendicular to another, anyone variable is less than another variable. Any Set P is a subset of Q, all these are examples of relations.

You will find the basics of the topic – relations, and functions and also its definition, types, and few solved examples on the same.

What is a Function?

It’s a set of ordered pair. It is also called as a point with two components as coordinates(x and y).

Example of an ordered pair is (6,-2), where 6 is x-coordinate and -2 is y-coordinate.

What is the Relation?

It is a subset of the Cartesian product. Or simply, a bunch of points(ordered pairs).

Example: {(-2, 1), (4, 3), (7, -3)}, it is written in set notation with curly brackets.

Relation in set notation:

There are other ways too to write the relation, apart from set notation through tables, plotting it on XY- axis or through mapping diagram.

Relations and Functions

Let us also learn about the Domain and Range of a given function:

Domain It is a collection of the first values in the ordered pairs(Set of all input (x) values).
Range It is a collection of the second values in the ordered pairs(Set of all output (y) values).

Example:

In the relation, {(-2, 3), {4, 5), (6, -5), (-2, 3)},

The domain is {-2, 4, 6} and Range is {-5, 3, 5}.

Note: Don’t consider duplicates while writing Domain and Range and also write it in increasing order.

How to convert a Relation into a function?

A special kind of relation(a set of ordered pairs) which follows a rule i.e every X-value should be associated to only one y-value is called a Function.

Let’s suppose, we have two relations given in table

A relation which is not a function A relation that is a function
Relation Relation1
As we can see duplications in X-values with different y-values, then this relation is not a function. As every value of X is different and is associated with only one value of y, this relation is a function.

Let’s Revise:

If there are any duplicates or repetitions in the X-value, the relation is not a function.

But there’s a twist here. Look at the following example:

Relation Example

Though X-values are getting repeated here, still it is a function because they are associating with the same values of Y.

The point (2, 6) is repeated here twice and (4, -8) is written thrice. We can rewrite it by writing a single copy of the repeated ordered pairs.

Now it is a function.

There are Different types of Relations

Those are as follows:

Types of Relations Empty
Universal
Identity
Inverse
Reflexive
Symmetric
Transitive

Empty Relation:

When there’s no element of set X is related or mapped to any element of X, then the relation R in A is an empty relation also called as void. I.e R= ∅.

For example,

if there are 100 mangoes in the fruit basket. There’s no possibility of finding a relation R of getting any apple in the basket. So, R is Void as it has 100 mangoes and no apples.

Universal relation:

R is a relation in a set, let’s say A is a universal Relation because, in this full relation, every element of A is related to every element of A. i.e R = A × A.

It’s a full relation as every element of Set A is in Set B.

Identity Relation.

If every element of set A is related to itself only, it is called Identity relation.

I={(A, A), Ɛ a}.

For Example,

When we throw a dice, the outcome we get is 36. I.e (1, 1) (1, 2), (1, 3)…..(6, 6). From these, if we consider the relation(1, 1), (2, 2), (3, 3) (4, 4) (5, 5) (6, 6), it is an identity relation.

Inverse Relation

If R is a relation from set A to set B i.e R Ɛ A X B. The relation R\(^{-1}\) = {(b,a):(a,b) Ɛ R}.

For Example,

If you throw two dice if R = {(1, 2) (2, 3)}, R\(^{-1}\)= {(2, 1) (3, 2)}. Here the domain is the Range R\(^{-1}\) and vice versa.

Reflexive Relation

A relation is a reflexive relation If every element of set A maps to itself. I.e for every a Ɛ A,(a, a) Ɛ R.

Symmetric Relation

A symmetric relation is a relation R on a set A if (a,b) Ɛ R then (b, a) Ɛ R, for all a &b Ɛ A.

Transitive Relation

If (a,b) Ɛ R, (b,c) Ɛ R, then (a,c) Ɛ R, for all a,b,c Ɛ A and this relation in set A is transitive.

Equivalence Relation

If and only if a relation is reflexive, symmetric and transitive, it is called an equivalence relation.

For example,

If we note down all the outcomes of throwing two dice, it would include reflexive, symmetry and transitive relations. that will be called an Equivalence relation.

Solve more examples of types of relations to understand the concept of reflexive symmetric and transitive relations.

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Practise This Question

If x=secθcosθ,y=sec10θcos10θ and (x2+4)(dydx)2=k(y2+4), then k is equal to