Download - Lecture 3 Functions
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Lecture 3
Functions
Chapter 2.3
Kenneth Rosen Book
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Functions
Def: Let A and B be sets. A function f (or more
completely, f : A B) is a rule that assigns to
each elementa A exactly one element f(a) B,
called the value off at a.
We also say that f : A B is a mapping from
domain A to codomain B.
f(a) is called the image of the element a, and the
element a is called a preimage of f(a).
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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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Some Function Terminology
If it is written that f:ApB, and f(a)=b(where aA &bB), then we say: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 R={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 functionis declaredto map all domain values into.
The range is theparticularset of values in
the codomain that the function actuallymaps 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 know fs codomain is:__________, and its range is ________.
Suppose the grades turn out all As and
Bs. Then the range offis _________, but its
codomain is __________________.
{A,B,C,D,E} unknown!
{A,B}
still {A,B,C,D,E}!
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Function Operator Example
, (plus,times) are binary operatorsoverR. (Normal addition & multiplication.)
Therefore, we can also add and multiplyfunctions f,g:RpR: (f g):RpR, where (f g)(x) = f(x) g(x)
(f g):RpR, where (f g)(x) = f(x) g(x)
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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 only1 pre-image.
Bipartite (2-part) graph representationsof functions that are (or not) one-to-one:
One-to-one
Not one-to-one
Not even a
function!
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Onto (Surjective) Functions
A function f:ApB is onto orsurjective ora surjection iff its range is equal to its
codomain (bB,aA: f(a)=b).
Think: An onto function maps the setA
onto (over, covering) the entiretyof the
set B, not just over a piece of it.
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Illustration of Onto
Some functions that are, or are not,ontotheir codomains:
Onto(but not 1-1)
Not Onto(or 1-1)
Both 1-1and onto
1-1 butnot onto
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Some more examples
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Bijections
A function fis said to be a one-to-one
correspondence, ora bijection, or
reversible, orinvertible, iff it is
both one-to-one and onto.
For bijections f:ApB, there exists aninverse of f, written f1:BpA, which is theunique function such that
(where IA is the identity function onA)AIff ! Q
1
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Composition of f and g
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Graphs of Functions
We can represent a function f:ApB as aset of ordered pairs {(a,f(a)) | aA}.
Note that a, there is only 1 pair(a,b).
For functions over numbers, we can
represent an ordered pair (x,y) as a point
on a plane. A function is then drawn as a curve (set of
points), with only one yfor eachx.
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A Couple of Key Functions
In discrete math, we will frequently use the
following two functions over real numbers:
The floorfunction -
:RZ, where -x (floor ofx) means the largest (most positive) integerex. I.e.,-x : max({iZ|ix}).
The ceilingfunction :RZ, where x(ceiling ofx) means the smallest (most
negative) integerux. -x : min({iZ|ix})
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Visualizing Floor& Ceiling
Real numbers fall to their floor or rise to
their 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 au0 anda
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Plots with floor/ceiling: Example
Plot of graph of function f(x) = -x/3:
x
f(x)
Set of points (x,f(x))
+3
2
+2
3
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Plots with floor/ceiling
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Review of 2.3 (Functions)
Function variables f,g,h,
Notations: f:ApB, 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 RpZ functions -x and x.