DISCRETE MATHEMATICS ILECTURES CHAPTER 7Dr. Adam AnthonySpring 2011

Some material adapted from lecture notes provided by Dr. Chungsim Han and Dr. Sam Lomonaco

FunctionsA function f from a set A to a set B is an assignment of elements in A to elements in B such that

Every item in A is assigned to something in BEvery item in A has only one assignment in B

We writef(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 writef: AB(note: Here, ““ has nothing to do with if… then)

Any function lacking these properties is not well-defined

FunctionsIf f:AB, we say that A is the domain of f and B is the codomain of f.

If f(a) = b, we say that b is the image of a and a is an inverse image of b.

The Complete inverse image of b B is the set {a A | f(a) = b}

there can be multiple items such that f(a) = b

The range of f:AB is the set of all images of elements of A.

We say that f:AB maps A to B.

FunctionsLet us take a look at the function f:PC withP = {Linda, Max, Kathy, Peter}C = {Boston, New York, Hong Kong, Moscow}f(Linda) = Moscowf(Max) = Bostonf(Kathy) = Hong Kongf(Peter) = New YorkHere, the range of f is C.

Functions

Let us re-specify f as follows:

f(Linda) = Moscowf(Max) = Bostonf(Kathy) = Hong Kongf(Peter) = Boston

Is f still a function?yes{Moscow, Boston, Hong Kong}

What is its range?

Functions

Other ways to represent f:

BostonPeter

Hong KongKathy

BostonMax

MoscowLinda

f(x)x Linda

Max

Kathy

Peter

Boston

New York

Hong KongMoscow

FunctionsIf the domain of our function f is large, it is convenient to specify f with a formula, e.g.:f:RR f(x) = 2xThis leads to:f(1) = 2f(3) = 6f(-3) = -6…

Exercise 1

X Y1 A2 B3 C4

a) Is f a function?b) What are the domain and

codomain of f?c) Write f as a set of ordered

pairsd) What is the range of f?e) Is 1 an inverse image of a?f) Is 2 an inverse image of b?g) What is the inverse image

of a?h) What is the inverse image

of b?i) What is the inverse image

of c?

f

Exercise 2—Is it a function?

1 2

3

2 4

No—2 is related to more than 1 element

Exercise 2—Is it a function?

1 2

3

2 4

Yes—Every element on the left is related to exactly one element on the right

Exercise 2—Is it a function?

1 2

3

2 4

No—2 is not related to anything.

Functions

Let f1 and f2 be functions from A to R.Then the sum and the product of f1 and f2 are also functions from A to R defined by:(f1 + f2)(x) = f1(x) + f2(x)(f1f2)(x) = f1(x) f2(x)

Example:f1(x) = 3x, f2(x) = x + 5(f1 + f2)(x) = f1(x) + f2(x) = 3x + x + 5 = 4x + 5(f1f2)(x) = f1(x) f2(x) = 3x (x + 5) = 3x2 + 15x

Example Function: Floor and Ceiling floor(x) = x

Largest integer y such that y x ceiling(x) = x

Smallest integer y such that y x Useful observation—for any real number

x: floor(x) x ceiling(x)

Exercise 3 Data stored on a computer disk or transmitted

over a network is typically represented as a string of bytes. Each byte is 8 bits. How many bytes are required to encode 100 bits of data?

In asynchronous transfer mode (a communications protocol used on backbone networks), data are organized into cells of 53 bytes. How many ATM cells can be transmitted in 1 minute over a connection that transmits data at the rate of 500 kilobits per second?

Exercise 4 Use floor and ceiling to write a new

definition for n div d and n mod d

Function Example: Logarithms Let b 1 be a positive real number. The

logarithm with base b is the logarithmic function

logb : R+ R and is defined by: logb x = the exponent to which b must be

raised to obtain x. Logbx = y by = x

Exercise 5 Compute the following:

log553

log91 log44 log24 log28 log2(1/2) log2(1/16)

Useful Logarithm Properties logb(xy) = logb(bsbt)

= logb(bs+t) = s + t= logbbs+ logbbt

= logbx + logby Similarly, logb(x/y) = logbx – logby

Base conversion: bxx

c

c

logloglogb

About log2

log2 comes up all the time in computing Basic fundamental problem: Given N items,

divide them into two groups. How many times can you divide the groups in half until there is at most one item in each group?

Another way of thinking of it: how many times can you divide an integer by 2 before you reach 0?

Any time you see this pattern (and we will!) remember that there is a logarithmic relationship here!

2-dimensional functions We can use cartesian products to define a

function in multiple dimensions The function f: A x B C maps pairs of elements

(a,b) to an element in C A binary operation is defined using a single

set: f:A x A Asum:Z x Z Z where sum(a,b) = a + b

A unary operation is similar, but does not use a cartesian product: f: A A

Exercise 6 Boolean Functions Given the set B = {T,F}, we can define a

boolean function with 2 inputs as: P:B x B B

Where the definition is given by a truth table:

Draw an arrow diagram for P

p q p q

T T TT F TF T TF F F

One-to-one and Onto functions.

Section 7.2

Function Images

We already know that the range of a function f:AB is the set of all images of elements aA.

If we only regard a subset SA, the set of all images of elements sS is called the image of S.

We denote the image of S by f(S):

f(S) = {f(s) | sS}

Functions

Let us look at the following well-known function:f(Linda) = Moscowf(Max) = Bostonf(Kathy) = Hong Kongf(Peter) = BostonWhat is the image of S = {Linda, Max} ?f(S) = {Moscow, Boston}What is the image of S = {Max, Peter} ?f(S) = {Boston}

Properties of Functions

A function f:AB is said to be one-to-one (or injective), if and only if

x, yA (f(x) = f(y) x = y)

In other words: f is one-to-one if and only if it does not map two distinct elements of A onto the same element of B.

Properties of FunctionsAnd again…f(Linda) = Moscowf(Max) = Bostonf(Kathy) = Hong Kongf(Peter) = BostonIs f one-to-one?

No, Max and Peter are mapped onto the same element of the image.

g(Linda) = Moscowg(Max) = Bostong(Kathy) = Hong Kongg(Peter) = New YorkIs g one-to-one?

Yes, each element is assigned a unique element of the image.

Properties of Functions

How can we prove that a function f is one-to-one?Whenever you want to prove something, first take a look at the relevant definition(s):x, yA (f(x) = f(y) x = y)Example:f:RRf(x) = x2

Disproof by counterexample:f(3) = f(-3), but 3 -3, so f is not one-to-one.

Properties of Functions… and yet another example:f:RRf(x) = 3xOne-to-one: x, yA (f(x) = f(y) x = y)To show: f(x) f(y) whenever x yx y 3x 3y f(x) f(y), so if x y, then f(x) f(y), that is, f is one-to-one.

Properties of Functions

A function f:AB with A,B R is called strictly increasing, if x,yA (x < y f(x) < f(y)),and strictly decreasing, ifx,yA (x < y f(x) > f(y)).

Obviously, a function that is either strictly increasing or strictly decreasing is one-to-one.

Properties of FunctionsA function f:AB is called onto, or surjective, if and only if for every element bB there is an element aA with f(a) = b.In other words, f is onto if and only if its range is its entire codomain.A function f: AB is a one-to-one correspondence, or a bijection, if and only if it is both one-to-one and onto.Obviously, if f is a bijection and A and B are finite sets, then |A| = |B|.

Properties of Functions

Examples:In the following examples, we use the arrow representation to illustrate functions f:AB.

In each example, the complete sets A and B are shown.

Properties of Functions

Is f injective?No.Is f surjective?No.Is f bijective?No.

Linda

Max

Kathy

Peter

Boston

New York

Hong KongMoscow

Properties of Functions

Is f injective?No.Is f surjective?Yes.Is f bijective?No.

Linda

Max

Kathy

Peter

Boston

New York

Hong KongMoscow

Paul

Properties of Functions

Is f injective?Yes.Is f surjective?No.Is f bijective?No.

Linda

Max

Kathy

Peter

Boston

New York

Hong KongMoscow

Lübeck

Properties of Functions

Is f injective?No! f is not evena function!

Linda

Max

Kathy

Peter

Boston

New York

Hong KongMoscow

Lübeck

Properties of Functions

Is f injective?Yes.Is f surjective?Yes.Is f bijective?Yes.

Linda

Max

Kathy

Peter

Boston

New York

Hong KongMoscow

LübeckHelena

Exercise 1—one to one and/or onto?

Let S by the set of all strings of 0’s and 1’s of any length. Let l:SZnonneg be the length function where l(s) = length of s.

The floor function

h:RR where h(x) = x2

Exercise 2 Let g:RR be the function defined by:

g(x) = 4x3 – 5for any real x. Prove that g is a one-to-one correspondence.

Two steps: prove it is one to one, then prove it is onto

Exercise 3 Show that the exponential function

E:RR+ given by E(x) = 2x is a one-to-one correspondence

Inversion

An interesting property of bijections is that they have an inverse function.

The inverse function of the bijection f:AB is the function f-1:BA with f-1(b) = a f(a) = b, for all b B

Inversion

Example:

f(Linda) = Moscowf(Max) = Bostonf(Kathy) = Hong Kongf(Peter) = Lübeckf(Helena) = New YorkClearly, f is bijective.

The inverse function f-1 is given by:f-1(Moscow) = Lindaf-1(Boston) = Maxf-1(Hong Kong) = Kathyf-1(Lübeck) = Peterf-1(New York) = HelenaInversion is only possible for bijections(= invertible functions)

Inversion

f-1:CP is no function, because it is not defined for all elements of C and assigns two images to the inverse image New York.

Linda

Max

Kathy

Peter

Boston

New York

Hong KongMoscow

LübeckHelena

f

f-1

Example 4 Find the inverse function of the

exponential function E:RR+ given by E(x) = 2x

Example 5 Find the inverse of g:RR, defined by:

g(x) = 4x3 – 5for any real x.

Composition of Functions

Section 7.3

CompositionThe composition of two functions g:AB and f:BC, denoted by fg, is defined by (fg)(a) = f(g(a))This means that • first, function g is applied to element aA, mapping it onto an element of B,• then, function f is applied to this element of B, mapping it onto an element of C.• Therefore, the composite function maps from A to C.

Composition

Example:

f(x) = 7x – 4, g(x) = 3x,f:RR, g:RR

(fg)(5) = f(g(5)) = f(15) = 105 – 4 = 101

(fg)(x) = f(g(x)) = f(3x) = 21x - 4

Exercise 1 Let X = {a,b,c} and Y = {1,2,3,4}. f:XY

and g:YX are functions defined by: f(a) = 2, f(b) = 1, f(c) = 3 g(1) = c, g(2) = b, g(3) = a, g(4) = b

Draw a combined arrow diagram to represent f, g and gf

What are the domain, codomain and range of g f?

Is g f a bijection (1-1 and onto)? If so, find its inverse function.

Exercise 2 Repeat Exercise 1, but now use f g.

Exercise 3 Let f:RR and g:RR be defined by:

f(x) = 2xg(x) = x2 + 1

for all reals x. Find g f and f g. Is g f = f g?

Identity Function

Composition of a function and its inverse:

(f-1f)(x) = f-1(f(x)) = x

The composition of a function and its inverse is the identity function i(x) = x.But this only works if f is a bijection!Useful equalities:

f i = fi f = f

Exercise 4 Let f:X Y and G:Y Z be functions.

If f and g are both one-to-one, then is g f one-to-one?

If f and g are both onto, is g f onto?

If g f is one-to-one, must both f and g be one-to-one?

If g f is onto, must both f and g be onto?