# Relations and Functions

## Relations and Functions:

This chapter of Relations and Functions forms the foundation of calculus in mathematics. But it can get confusing for many to understand the difference between the two. In this article, we will be talking about both these concepts in detail.

Just like the relations we all have in our daily lives, there are relations in the world of algebra too. In our lives, relations are more about people like brothers, sisters, friends or students and teachers. In math, there are some relations like a line is parallel or perpendicular to another, so any one variable is greater or less than another variable. Any Set A is a subset of B, and these are all examples of relations. Relations and functions are closely tied with each other, hence the need to study them together.

One thing that is very common while studying relations is the fact that it requires two different objects to link two different objects. Find out more about relations and functions in the space that follows.

## What are Relations?

For understanding relations, we need the basic knowledge of sets. A Set can be explained as a collection of well-defined objects of a particular kind. For instance, it could be a set of outcomes of dice or a set of the English alphabet.

The study of relations is always done between two sets. If there are two non-void (or null/empty) sets A and B, then the relation R from set A to set B can be represented by aRb, wherein a is the set of elements that belong to set A while b belongs to set B.

The relation from a set A to a set B is the subset of the Cartesian product of A and B i.e. subset of A x B. Relation, in other terms, can also be classified as a collection of ordered pairs (a, b) where a belongs to the elements from set A and b from set B and the relation is from A to B but not vice versa.

Example

Let’s consider a set A with elements {1, 2, 3} and set B with elements {2, 4, 6}.

In this case, the relation between Set A and B from A to B will be set of any combinations from Set A to set B.

If you look at the above diagram, you’ll see that the Relation from A to B i.e. will be set of {(1,4). (1,2), (3,4), (3,2)}. This relation is a subset of the Cartesian product of two sets A X B.

Let’s talk about another instance where, set A = {1, 2, 3} and set B = {1, 2, 3, 4, 5, 6, 7, 8, 9}.

If the Relation between A and B is as: elements of B is the squares of elements of set A, then the relation is written in the form of sets as:

R = {(a,b): where b is square of a and a ∈ A & b ∈ B}

Then R = {(1,1), (2,4), (3,9)}

Just like sets, relations can also be written algebraically either by the Roster method or by the Set-builder method.

We can also define Relation as a linear operation that is the reason behind establishing relationships between the elements of two sets according to some definite rule of relationship.

## What are Functions?

Functions can be defined as a special class of relation or we can simply say that they are special types of relations. Function is a very important concept in mathematics as in our real life, every situation is solved and analysed by writing first its mathematical equation or function.

A function can also be thought of as a machine that gives a unique output for each input that is fed into it. But, as we know, every machine is specifically made for certain inputs such as a washing machine is meant for washing clothes and not the wood. Likewise, all kinds of functions are made for certain inputs that are called its domain and corresponding outputs are called Range.

Assuming A and B are two sets and correspondence ‘f’ associates to each element of A to a unique element in B, then f is called a Function or Mapping from A to B. It is denoted by the symbol:

This can be read as ‘f is a function from A to B’ or f maps A to B.

Let’s say that an element a ∈ A is connected with an element b ∈ B, then b would be called ‘the f image of a’ or ‘image of a under f’ or ‘the value of the function f at a’. Also, in this case, a is called the pre–image of b or argument of b under the function f. We would be writing it as:

f:(a, b) or f:a → b or b = f(a)

A relation f from a set A to a set B is known as the function if it adheres to the following conditions:

• All the elements of A should be mapped to the elements of B. Which implies that there shouldn’t be any element in A which is being unmapped with B.

i.e. a, (a, f(a)) f, where, a is the elements of set A

• Elements of set A should be uniquely mapped with the elements of set B.

i.e. if (a, b) f & (a, c) f, b = c

Therefore, the ordered pairs of must match the property that each element of A appears in some ordered pair and no two ordered pairs have the same first elements.

## What is the Domain of a Function?

When it comes to a relation from set A to set B i.e. aRb, all the elements that are a part of set A are called as the domain of the relation R. Similarly, the elements of set B are known as the co-domain of the relation R.

In this context, a range can be defined as the set of all second elements from the ordered pairs (a, b) in the relation aRb.

Domain of f = {a | a ∈ A, (a, f(a)) ∈ f }

Range of f = {f (a) |a ∈ A, f(a) ∈ B, (a, f (a)) ∈ f}

So, in terms of relation aRb, domain is basically the input to relation R, the co-domain is the possible output and the range is the actual output.

In the same way for a function, f: A → B, elements of set A are the inputs and B is the set of possible outputs. But in this case, the second elements of all the ordered pair of f(A, B) would be the actual outputs.

You need to note that the range is a subset of co-domain. When the rule of the function is given, then the domain of the function is the set of those real numbers, wherein the function is defined. But for the continuous function, the interval from minimum to maximum value of a function gives the range.

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