Functions1 Elementary Discrete Mathematics Jim Skon.

Functions 1

FunctionsElementary Discrete Mathematics

Jim Skon

Functions 2

Functions

Definition: A function consists of three things:1. A non empty set A, called the domain of the

function

2. A non empty set B, called the codomain of the function

3. A rule that assigns to each element of A one and only one, element of B

Functions 3

Function Notations

Use letters such as ƒ, g, and h to denote functions.

ƒ:A B means ƒ is a function with domain A and codomain B

ƒ:A B is read: "ƒ is a function from A to B”

x f(x)

A Bf

Functions 4

Function Notations

If ƒ:A B, and a A and b B, and a is assigned to b by the function ƒ, then we say ƒ(a) = b

If ƒ(a) = b, then the element ƒ(a) or b is called the value of ƒ at a, or the image of a under the function ƒ.

a f(a)=b

A Bf

Functions 5

Function Example Let A = {1,2,3} and B = {a,b,c}

Let ƒ(1) = bƒ(2) = cƒ(3) = a

c is the image of 2 under the function ƒ. The image f(A) of function f is B = {a, b,

c}1

2

3

b

c

a

A B

ƒ

Functions 6

Functions

In general, element a A maps to element ƒ(a) B.

a

A B

ƒ

ƒ(a)

Functions 7

Functions

From rule 3 of the definition, elements of the domain can map to at most one element of the codomain.

Multiple elements of the domain may map to the same element of the range:

1

2

3

b

c

a

A B

ƒ

function

Functions 8

Functions

The domain may not map to multiple elements of the range:

This is called the uniqueness condition of functions

1

2

3

b

c

a

A B

ƒ

not a function

Functions 9

Function

.Formal Definition: Let A and B be sets. A function ƒ:A B is a subset of the Cartesian

product A B, which satisfies the uniqueness condition that, for all (a1, b1) ƒ and (a2, b2) ƒ, if a1 = a2, then b1 = b2.

Functions 10

Function Examples:

Consider again A = {1,2,3} and B = {a,b,c}

Let ƒ(1) = bƒ(2) = cƒ(3) = a

ƒ = {(1, b), (2, c), (3, a) }

Functions 11

Function Examples

Consider ƒ:R R where R is reals

Let ƒ(x) = x2. Alternately: ƒ = {(x, x2) | x R} Then ƒ:R R

Functions 12

Function Examples

Let ƒ:N N where

ƒ(n) = n + 1 for all n N Alternately :ƒ = {(n, n+1) | n N }

Functions 13

Function Examples

Let S be a finite non-empty set. We may define the function:

ƒ:P(S) N as ƒ(A) = |A|.

Alternately: ƒ = {(a, n) | a P(S) n N |a| = n}

This is the set size function.

Functions 14

Function Examples

Consider the function ƒ:N N N

N N are pairs of natural numbers. Let ƒ(x,y) = x2 + y ƒ = {((x,y), x2 + y)}

Functions 15

Function Examples

Consider a function ƒ:P(S) P(S) P(S)

where S = {1, 2, 3, ..., 10} Let ƒ(A,B) = A B ƒ = {((A, B), A B) | A S B S }

This is the union function

Functions 16

Function Examples Consider in general n-ary functions, which are of the form

ƒ:A1 A2 ... An B.

These are called n-ary functions or functions of n variables, and are written:

ƒ(a1, a2, ..., an) = b

Functions 17

Function Examples

Consider a function ƒ:N N N N I

Let ƒ(w, x, y, z) = 2w + 3(xy) - 4z

ƒ = {( (w, x, y, z), 2w + 3(xy) - 4z) }

Functions 18

Function Examples

Consider a function ƒ:A B NA = {x | x is a MVNC basketball player}

B = {x | x is a MVNC basketball game (date)}

Let ƒ(x, y) = points scored by player x in game y

Functions 19

Function Examples

Consider a function ƒ:A B NA = {x | x is a first names}

B = {x | x is a last name}

Let ƒ(x, y) = student x y’s box number.

Not a function! Why?

Functions 20

Function Examples Consider function ƒ:R I where:

ƒ(x) = Largest integer less than or equal to x.

ƒ(x) = xcalled the floor function.

Consider function ƒ:R I where:ƒ(x) = Least integer greater than or equal to x.

ƒ(x) = xcalled the ceiling function.

Functions 21

Function Range

Range Definition: Let ƒ:A B be a function from A (domain) to

B (codomain). The range of ƒ is the set of all elements of B that are mapped to by some element of A, i.e.

range(ƒ) = {b B | b = ƒ(a) for some a A}

In other words, the range of ƒ is the subset of B which the function actually maps to.

Functions 22

Surjective (onto) Function

Let B be the codomain of function ƒ. If range(ƒ) = B, then we say that the function

is onto BA function ƒ:A B is surjective if it is onto

B.In other words, a function is surjective if

every element in the codomain is mapped to.

Functions 23

Surjective FunctionA BRange(f)

A BRange(g)

g

f

Not onto

Ontof:AB

g:AB

Functions 24

Surjective Function

Which of the previous examples are surjective?

Functions 25

Surjective Function

Formally ƒ is surjective if and only if

b a ƒ (a) = b.or

b a (a, b) ƒ

Functions 26

Injective FunctionDefinition: Let ƒ:A B. If no two different elements of

A are assigned to the same element of B by the function ƒ, the function is one-to-one.

More formally if a1 A:a2 A: ƒ(a1) = ƒ(a2) a1 = a2 Then the function is one-to-one.

Contrapositively:a1 A:a2 A:a1 a2 ƒ(a1) ƒ(a2)

Functions 27

Injective FunctionA B

g

f

Not one to one

One to onef:AB

g:AB

f(a1)a1

a2

a3

a4

a5

a6

f(a2)f(a3)f(a4)f(a5)f(a6)

A B

f(a1)a1

a2

a3

a4

a5

a6

f(a2)

f(a3)= f(a4)

f(a5)f(a6)

f(a7)

f(a7)

Functions 28

Injective functions:

If a function is one-to-one then it is injective.

Which of the previous examples are injective?

Functions 29

Example

Let ƒ:N N be defined by: ƒ(n) = n2

Is this surjective? injective?

Let ƒ:I N be defined by: ƒ(n) = n2

Is this surjective? injective?

Functions 30

Bijective Function

Definition - bijective function

If a function is both surjective and injective then it is bijective.

A bijective function is onto and one-to-one.

A bijective function is simply a one-to-one correspondence

Functions 31

Bijective Function

A function ƒ:A B is bijective if and only if

b B:!a A:ƒ(a) = b

Which of the previous examples are bijective?

Functions 32

Function Composition

Let ƒ:A B and g:B CWe can now define a new function, g f, by the

formula:

(gf)(a) = g(ƒ(a))

This is called the composition function of g with ƒ.

Functions 33

Function Composition

a ƒ(a)

ƒ g

gfA B C

g(ƒ(a)) = gf(a)

Functions 34

Function Composition ExampleLet A = {x, y, z} B = {2, 4, 6, 8}

C = {}Let ƒ:A B be defined by:

ƒ(x) = 2, ƒ(y) = 8, ƒ(z) = 4

Let g:B C be defined by:g(2) = , g(4) = , g(6) = g(8) =

Then g ƒ:A C is the function:(g ƒ)(x) = (g( ƒ(x)) = g(2) = (g ƒ)(y) = (g( ƒ(y)) = g(8) = (g ƒ)(z) = (g( ƒ(z)) = g(4) =

Functions 35

Function Composition ExampleLet ƒ:R R be defined by:

ƒ(x) = 2x2 + 4Let g:R R be defined by:

g(x) = 3x - 1Then g ƒ:(x) = g( ƒ(x))

= g(2x2 + 4)= 3(2x2 + 4) - 1= 6x2 + 12 - 1= 6x2 + 11

What is ƒ g:(x)?

Functions 36

Function Composition Example

Let S be a finite set and x S. We can define:

ƒ:P(S) P(S {x})as the function:

ƒ(T) = T {x}, where T S (or T P(S) )

Let g:P(S {x}) N be the function:

g(V) = |V|, where V S {x} (or V P(S {x}) )

Functions 37

Function Composition Example

Then the composition:

g ƒ:P(S) N

is given by

(g ƒ)(T) = (g( ƒ(T))

= g(T {x})

= |T {x}|

= |T| + 1

Functions 38

Function Composition

In general the composition of functions is not communitive,

e.g. ƒ g g ƒ. In fact, if ƒ g is possible, g ƒ is usually

not possible!

Functions 39

Function Composition

For g ƒ to be possible, f must have a codomain which is a subset of the domain of g.

If ƒ:A B and g:C D, then B C.A CB

g

g:CD

f

D

f:AB

Functions 40

Function Composition

Likewise, for ƒ g to be possible, g must have a codomain which is a subset of the domain of f, e.g.

If ƒ:A B and g:C D, then D A.C AD

f

g:CD

g

B

f:AB

Functions 41

Function Composition

Thus for both ƒ g & g ƒ to exist, B C and D A

A CB

g

g:CD

f

D

f:AB

Functions 42

INVERSE of Functions

If ƒ:A B is a bijection, then it is possible to define a function g:B A with the property: If ƒ(a) = b then g(b) = a

A B

g:BA

f:AB

Functions 43

INVERSE of Functions

Such a function g is called the inverse function of ƒ.

It is denoted by the symbol ƒ-1.

A B

f-1:BA

f:AB

If ƒ(a) = b then f-1(b) = a

Functions 44

Example

Let A = {1,2,3} and B = {a,b,c}.Let ƒ:A B be defined by: ƒ(1) = c, ƒ(2) = a, ƒ(3) = bThen the inverse ƒ-1:B A is defined by: ƒ-1(a) = 2, ƒ-1(b) = 3, ƒ-1(c) =

123

cab

A B

f123

cab

A B

f-1

Functions 45

INVERSE of Functions

The the function ƒ:A B has an inverse ƒ-1:B A if and only if it is bijective.

WHY??

Functions 46

INVERSE of Functions

Note that in general

ƒ-1 ƒ(a) = a, for all a in the domain of ƒ

ƒ ƒ-1(b) = b, for all b in the codomain of ƒ

Functions 47

Function Images

Consider a function: :NN, where (x) =2xThe range of the function is: {0, 2, 4, 6, 8, ... }

Functions 48

Function Images

We can also consider the image of the function over a subset of the domain.

Let A = {2, 3, 4, 5, 6}.

(A) is the the image of function over set A, which is:

(A) = {4, 6, 8, 10, 12}If B = {x | 4 < x 10} then

(B) = {10, 12, 14, 16, 18, 20}

Functions 49

Function Images

Image of a function - the elements mapped to over a given subset

Consider ƒ:R R , f(x) = 2x f(R) = Range(R) = R (The range of f). f(I) = I Let A = {1, 4, 6, 9). Then f(A) = {2, 8, 12, 18} f(N) = ? f(R-) = I

Functions 50

Function Images

Definition: Let be a function from set A to set B and let S

be a subset of A (e.g. S A). The image of S is the subset of B that consists

of the images of the elements of S. The image of S is denoted (S), thus:

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

Functions 51

Function Images

Consider::RR, where (x) = (x+1)/2If S = {1, 3, 5, 7, 9}, what is (S)?If S = {x | 3 x 6}, what is (S)?What is (N)?What if (Z)?

Functions 52

Example

Consider 12 - 14 on page 70