Lec 04 function

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LECTURE 04

Transcript of Lec 04 function

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Textbook

Discrete Mathematics & Its Applications - Kenneth H. Rosen

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Topics to be covered todayFunction

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FunctionLet A and B be nonempty sets

A Function f from A to B Is an assignment of exactly one element of B to each element of A

We write f(a) = b If b is the unique element of B Assigned by the function f to the element a of A

 If f is a function from A to B

We write f: A B

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FunctionFunctions are specified in many different ways

Sometimes we explicitly state the assignments

As in following Figure

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Domain, CoDomain, Range, Image, PreImageIf f is a function from A to B

A is the domain of f B is the co-domain of f

 If f(a) = b b is the image of a a is a pre-image of b

 The range of f: Set of all images of elements of A We define a function by it’s

DomainCo-domainMapping of elements of the domain to elements in the co-domain

 

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Domain, CoDomain, Range, Image, PreImage2 Functions are Equal - When they have Same domain Same co-domain Map elements of their domain to the same elements in their co-domain Note: If we change either the Domain or Co-Domain of a function

Then we obtain a different function If we change the Mapping of Elements

Then we also obtain a different function

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Domain, CoDomain, Range, Image, PreImage

What are the Domain, Codomain & Range of the function that assigns grades to students?

Solution:

Let G be the function that assigns a grade to a studentFor example: G(Adams) = A

Domain of G = Set {Adams, Chou, Goodfriend, Rodriguez, Stevens} Codomain of G = Set {A, B, C, D, F}Range of G = Set {A, B, C, F}

Because each grade except D is assigned to some student

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Domain, CoDomain, Range, Image, PreImageLet R be the relation consisting of ordered pairs (Abdul, 22), (Brenda, 24), (Carla, 21), (Desire, 22), (Eddie, 24), (Felicia, 22)

Where each pair consists of: A Graduate student & Age of the studentWhat is the function that this relation determines?

Solution:

This relation defines the function f, Where f(Abdul)=22, f(Brenda)=24, f(Carla)=21, f(Desire)=22, f(Eddie)=24, f(Felicia)=22

Domain: Set {Abdul, Brenda, Carla, Desire, Eddie, Felicia}

Codomain: Set of positive integers To make sure that the Codomain contains all possible ages of students

Range: Set {21, 22, 24}

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FunctionLet f be the function

That assigns the last 2 bits of a bit string of length 2 or greater to that string

For Example: f(11010) = 10

Domain of f:

Set of all bit strings of length 2 or greater

Codomain & Range of f:

Set {00, 01, 10, 11}

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Function

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FunctionLet f be a function from Set A to Set B Let S be a subset of A

The image of S under the function f:Subset of B that consists of the images of the elements of S

We denote the image of S by f(S)

Let A = {a, b, c, d, e} & B = {l, 2, 3, 4} With f(a) = 2, f(b) = 1, f(c) = 4, f(d) = 1, f(e) = 1

The image of the subset S = {b, c, d}: Set f(S) = {1, 4}

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Classification of Function3 Types of Function:

One-to-One

Onto

One-to-One Correspondence

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Function – One to OneSome functions never assign the same value

To 2 different domain elements

These functions are said to be One-to-One

A function f is said to be one-to-one

If and only if f(a) = f(b) implies that a = b For all a & b in the domain of f

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Function – One to OneDetermine whether the function f

From {a, b, c, d} to {l, 2, 3, 4, 5} With f(a) = 4, f(b) = 5, f(c) = 1, f(d) = 3 is one-to-one

Solution: The function f is one-to-one Because f takes on different values at 4 elements of its domain

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Function - OntoFor some functions Range & Co-domain are equal

Every member of the co-domain Is the image of some element of the domain

Functions with this property are called onto Functions

A function f from A to B is called onto

If and only if for every element b of set B There is an element a of set A with f(a) = b

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Function - OntoLet f be the function from {a, b, c, d} to {l, 2, 3}

Defined by f(a) = 3, f(b) = 2, f(c) = 1, f(d) = 3Is f an onto function?

Solution: Because all 3 elements of Codomain are images of elements in domain

We say that f is onto

This is illustrated in following Figure

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Function - One to One CorrespondenceThe function f is a One-to-One Correspondence or Bijection

If it is both one-to-one and onto

Example:

Let f be the function from {a, b, c, d} to {l, 2, 3, 4} With f(a) = 4, f(b) = 2, f(c) = 1, f(d) = 3

Is f a Bijection?

Solution: The function f is both One-to-One & onto• It is one-to-one Because no 2 values in the domain are assigned the same value• It is onto Because all 4 elements of Codomain: Are images of elements in Domain

So f is a Bijection

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FunctionFollowing Figure displays 4 functions Where

• First is one-to-one but not onto• Second is onto but not one-to-one• Third is both one-to-one and onto• Fourth is neither one-to-one nor onto• Fifth is not a function, because it sends an element to 2 different elements

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Identity FunctionLet A be a set

The identity function on A is the function iA : A AWhere iA(x) = x

In other words

The identity function iA is the function That assigns each element to itself

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Inverse FunctionLet f be a One-to-One Correspondence from the set A to the set B

Inverse Function of f:Function that assigns

To an element b belonging to B Unique element a in A

Such that f(a) = b

The inverse function of f is denoted by f-1

Hencef-1(b) = a When f(a) = b

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Invertible FunctionA one-to-one correspondence is called Invertible

Because we can define an inverse of this function

A function is not invertible: If it is not a one-to-one correspondence Because - Inverse of such a function does not exist

Example:Let f be the function from {a, b, c} to {1, 2, 3}

Such that f(a) = 2, f(b) = 3, f(c) = 1Is f invertible, and if it is, what is its inverse?

Solution: The function f is invertible because it is a one-to-one correspondenceThe inverse function f-1 reverses the correspondence given by f

So: f-1(1) = c, f-1(2) = a, f-1(3) = b

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Composition of Functions

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Composition of Functions

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Composition of Functions