functions note
TRANSCRIPT
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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.