# Discrete Mathematics Functions - Discrete Mathematics

## What are Discrete Mathematics Functions?

One element of a related set is assigned to each element of the set by using a Function. For representing a computational complexity of algorithms, for counting objects, for studying the sequences and strings, and for naming some of them, functions are used.

## Define Discrete Mathematics Function

The relationship from the elements of one set X to elements of another set Y is defined as function or mapping, which is represented as f:X→Y. For the function ‘f’, X is the domain or pre-image and Y is the codomain of image. Function ‘f’ is a relation on X and Y such that for each x∈X, there exists a unique y∈Y such that (x,y)∈R. A function can be one to one or many to one.

### What is Injective / One-to-one function?

A function f:A→B is injective or one-to-one function if for every b∈B, there exists at most one a∈A such that f(s)=t.

This means a function f is injective if Example is not injective as (−x)2=x2

### What is Surjective / Onto function?

A function f:A→B is surjective (onto) if the image of f equals its range. Equivalently, for every b∈B, there exists some a∈A such that f(a)=b. This means that for any y in B, there exists some x in A such that y=f(x).

Example ### What is Bijective / One-to-one Correspondent?

If a function f is both injective and surjective, then the function f:A→B is bijective or one-to-one correspondent.

Problem

Prove that a function f:R→R defined by f(x)=2x–3 is a bijective function.

Explanation – This function is proved by using both injective and surjective functions. Here, 2x–3=y

So, x=(y+5)/3 which belongs to R and f(x)=y.

Hence, f is surjective.

Since f is both surjective and injective, it is said tha f is bijective.

## What is Inverse of a Function?

The inverse of a one-to-one corresponding function f:A→B, is the function g:B→A, holding the following property −

f(x)=y⇔g(y)=x

The function f is called invertible, if its inverse function g exists.

Example

• A Function f:Z→Z,f(x)=x+5, is invertible since it has the inverse function g:Z→Z,g(x)=x−5.
• A Function f:Z→Z,f(x)=x2 is not invertiable since this is not one-to-one as (−x)2=x2.

## How the Functions can be composed?

Two functions f:A→B and g:B→C can be composed to give a composition gof. This is a function from A to C defined by (gof)(x)=g(f(x))

### Example

Let f(x)=x+2 and g(x)=2x+1, find (fog)(x) and (gof)(x).

### Solution

(fog)(x)=f(g(x))=f(2x+1)=2x+1+2=2x+3

(gof)(x)=g(f(x))=g(x+2)=2(x+2)+1=2x+5

Hence, (fog)(x)≠(gof)(x)

Some Facts about Composition

• The function (gof) is one-to-one if f and g are one-to-one.
• The function (gof) is onto if f and g are onto.
• The composition of functions always goes along with the associative property and does not hold the cumulative property.

Discrete Mathematics Topics