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.

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