functions note

7/30/2019 Functions Note

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Functions


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On to Functions

From calculus, you are familiar with the

concept of a real-valued functionf,

which assigns to each numberxRa

particular valuey=f(x), whereyR.But, the notion of a function can also be

naturally generalized to the concept of

assigning elements ofany set to elementsofany set.


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Function: Formal Definition

For any setsA,B, we say that afunction f

from (or mapping) A to B (f:AB) is aparticular assignment of exactly one

elementf(x)B to each elementxA.Some further generalizations of this idea:

Apartial(non-total) functionfassignszero or

one elements ofB to each elementxA.Functions ofn arguments; relations (ch. 6).


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Graphical Representations

Functions can be represented graphically in

several ways:

AB

a b

f

f

x

y

PlotBipartite Graph

Like Venn diagrams

A B


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Functions Weve Seen So Far

Aproposition can be viewed as a function

from situations to truth values {T,F}

A logic system calledsituation theory.

p=It is raining.;s=our situation here,now

p(s){T,F}.

Apropositional operatorcan be viewed asa function from ordered pairs of truth

values to truth values: ((F,T)) = T.


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More functions so far

Apredicate can be viewed as a function

from objects topropositions (or truth

values): P: is 7 feet tall;

P(Mike) = Mike is 7 feet tall. = False.A bit string B of length n can be viewed as a

function from the numbers {1,,n}

(bitpositions) to the bits {0,1}.E.g., B=101 B(3)=1.


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Aset Sover universe Ucan be viewed as a

function from the elements ofUto

{T, F}, saying for each element ofU

whether it is in S. S={3}; S(0)=F, S(3)=T.Aset operatorsuch as ,, can be

viewed as a function from pairs of sets

to sets.Example: (({1,3},{3,4})) = {3}

Still More Functions


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A Neat Trick

Sometimes we write YXto denote the setFofallpossible functionsf:XY.

This notation is especially appropriate,because for finiteX, Y, |F| = |Y||X|.

If we use representations F0, T1,2

:

{0

,1

}={F

,T

}, then a subset T

Sis just afunction from Sto 2, so the power set ofS(set of all such fns.) is 2S in this notation.


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Some Function Terminology

Iff:AB, andf(a)=b (where aA & bB),then:

A is the domain off.

B is the codomain off.

b is the image ofa underf.

a is apre-image ofb underf.

In general, b may have more than 1 pre-image.The range RB offis {b | a f(a)=b }.


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Range versus Codomain

The range of a function might notbe its

whole codomain.

The codomain is the set that the function is

declaredto map all domain values into.

The range is theparticularset of values in

the codomain that the function actually

maps elements of the domain to.


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Range vs. Codomain - Example

Suppose I declare to you that: fis a

function mapping students in this class to

the set of grades {A,B,C,D,E}.

At this point, you knowfs codomain is:

__________, and its range is ________.

Suppose the grades turn out all As and Bs.

Then the range offis _________, but its

codomain is __________________.


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Operators (general definition)

An n-ary operatorover the set Sis any

function from the set of ordered n-tuples of

elements ofS, to Sitself.

E.g., ifS={T,F}, can be seen as a unaryoperator, and , are binary operators on S.

Another example: and are binaryoperators on the set of all sets.


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Constructing Function Operators

If (dot) is any operator overB, then wecan extend to also denote an operator overfunctionsf:AB.

E.g.: Given any binary operator:BBB,and functionsf,g:AB, we define(fg):AB to be the function defined by:

aA, (fg)(a) =f(a)g(a).


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Function Operator Example

, (plus,times) are binary operatorsoverR. (Normal addition & multiplication.)

Therefore, we can also add and multiply

functions f,g:RR:(fg):RR, where (fg)(x) =f(x) g(x)

(fg):RR, where (fg)(x) =f(x) g(x)


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Function Composition Operator

For functionsg:AB andf:BC, there is aspecial operator called compose ().

It composes (creates) a new function out off,g

by applyingfto the result ofg.(fg):AC, where (fg)(a) =f(g(a)).

Noteg(a)B, sof(g(a)) is defined and C.

Note that (like Cartesian , but unlike +,,)is non-commuting. (Generally,fggf.)


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Images of Sets under Functions

Givenf:AB, and SA,

The image ofSunderfis simply the set of

all images (underf) of the elements ofS.

f(S) : {f(s) |sS}: {b | sS:f(s)=b}.

Note the range offcan be defined as simply

the image (underf) offs domain!


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One-to-One Functions

A function is one-to-one (1-1), orinjective, oran injection,

iff every element of its range has only 1 pre-image.

Formally: givenf:AB,

x is injective : (x,y:xy f(x)f(y)).

Only one element of the domain is mapped to any given

one element of the range.

Domain & range have same cardinality. What about codomain?

Each element of the domain is injected into a differentelement of the range.

Compare each dose of vaccine is injected into a different patient.


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One-to-One Illustration

Bipartite (2-part) graph representations of

functions that are (or not) one-to-one:

One-to-one

Not one-to-one

Not even a

function!


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Sufficient Conditions for 1-1ness

For functionsfover numbers,

fisstrictly (ormonotonically) increasingiff

x>y f(x)>f(y) for allx,y in domain;

fisstrictly (ormonotonically) decreasingiffx>y f(x)


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Onto (Surjective) Functions

A functionf:AB is onto orsurjective orasurjection iff its range is equal to its

codomain (bB, aA:f(a)=b).An onto function maps the setA onto (over,

covering) the entirety of the setB, not just

over a piece of it.E.g., for domain & codomain R, x3 is onto,

whereasx2 isnt. (Why not?)


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Illustration of Onto

Some functions that are or are not onto theircodomains:

Onto

(but not 1-1)

Not Onto

(or 1-1)

Both 1-1

and onto

1-1 but

not onto


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Bijections

A functionfis a one-to-one correspondence,ora bijection, orreversible, orinvertible, iff

it is both one-to-one and onto.

For bijectionsf:AB, there exists aninverse of f, writtenf1:BA, which is theunique function such that

(the identity function)

Iff 1


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The Identity Function

For any domainA, the identity functionI:AA (variously written,IA, 1, 1A) is theunique function such that aA:I(a)=a.

Some identity functions youve seen:ing 0, ing by 1, ing with T, ing with F,

ing with , ing with U.

Note that the identity function is both one-

to-one and onto (bijective).


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The identity function:

Identity Function Illustrations

Domain and range x

y


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Graphs of Functions

We can represent a functionf:AB as a setof ordered pairs {(a,f(a)) | aA}.

Note that a, there is only 1 pair (a,f(a)).Later (ch.6): relations loosen this restriction.

For functions over numbers, we can

represent an ordered pair (x,y) as a point ona plane. A function is then drawn as a curve

(set of points) with only oney for eachx.


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Comment About Representations

You can represent any type of discretestructure (propositions, bit-strings, numbers,sets, ordered pairs, functions) in terms ofvirtually any of the other structures (or

some combination thereof).Probably none of these structures is truly

more fundamental than the others (whateverthat would mean). However, strings, logic,

and sets are often used asthe foundation for all else. E.g. in


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A Couple of Key Functions

In discrete math, we will frequently use thefollowing functions over real numbers:

x (floor ofx) is the largest (most positive)

integerx.x (ceiling ofx) is the smallest (most

negative) integerx.


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Visualizing Floor & Ceiling

Real numbers fall to their floor or rise totheir ceiling.

Note that ifxZ,

x x &x x

Note that ifxZ,

x = x =x.

0

1

1

2

3

2

3

...

.

. .

. . .

1.6

1.6=2

1.4= 2

1.4

1.4= 1

1.6=1

3

3=3= 3


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Plots with floor/ceiling

Note that forf(x)=x, the graph offincludes thepoint (a, 0) for all values ofa such that a0 anda


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Plots with floor/ceiling: Example

Plot of graph of functionf(x) = x/3:

x

f(x)

Set of points (x,f(x))

+3

2

+2

3


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Review of2.3 (Functions)

Function variablesf,g, h,

Notations:f:AB, f(a),f(A).

Terms: image, preimage, domain, codomain,

range, one-to-one, onto, strictly (in/de)creasing,bijective, inverse, composition.

Function unary operatorf1,

binary operators , , etc., and.

The RZ functions x and x.

functions note

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