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Chapter 7, Functions

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Function terminology

A relationship between elements of two sets such that no element of the first set is related to more than one element of the second set

Domain: the set which contains the values to which the function is applied

Codomain: the set which contains the possible values (results) of the function

Range (or image): the set of actual values produced when applying the function to the values of the domain

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More function terminology f: X Y

– f is the function name– X is the domain– Y is the co-domain– x X y Y f sends x to y– f(x) = y f of x; the value of f at x ; the image of x under f

A total function is a relationship between elements of the domain and elements of the co-domain where each and every element of the domain relates to one and only one value in the co-domain

A partial function does not need to map every element of the domain

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Formal definitions

The range of f is {y Y | (x X)[f(x) = y]}–where X is the domain and Y is the co-domain

The inverse image of y Y is{x X | f(x) = y}–the set of things in the domain X that map to y

Arrow diagramsDetermining if something is a function using an arrow diagram

Equality of functions

( functions f,g with the same domain X and codomain Y) [f = g iff (x X)[f(x) = g(x)] ]

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Discrete StructuresCMSC 250Lecture 38

April 30, 2008

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Types of functions

F:X Y is a one-to-one (or injective) function iff(x1,x2 X)[F(x1) = F(x2) x1 = x2], or alternatively

(x1,x2 X)[x1 x2 F(x1) F(x2)]

F: X Y is not a one-to-one function iff(x1,x2 X)[(F(x1) = F(x2)) ^ (x1 x2)]

F: X Y is an onto (or surjective) function iff(y Y)(x X)[F(x) = y]

F: X Y is not an onto function iff(y Y)(x X)[F(x) y]

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Proving functions one-to-one and onto

f: R R f(x) = 3x 4

Prove or give a counterexample that f is one-to-one– recall the definition (one of two definitions) of one-to-one is

Prove or give a counterexample that f is onto– recall the definition of onto is

])()([),( 212121 xxxfxfRxx

])([)()( yxfRxRy

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One-to-one correspondence or bijection

F: X Y is bijective iff F: X Y is one-to-one and onto

If F: X Y is bijective then it has an inverse function

])()([)(

])()([)(

][)(

1

1

1

yxFxyFYy

xyFyxFXx

XYF

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Proving something is a bijection

F: Q Q F(x) = 5x + 1/2– prove it is one-to-one– prove it is onto– then it is a bijection– so it has an inverse function

• find F1

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The pigeonhole principle

Basic form:

A function from one finite set to a smaller finite set cannot be one-to-one; there must be at least two elements in the domain that have the same image in the codomain.

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Examples Using this class as the domain:

– must two people share a birth month? – must two people share a birthday?

Let A = {1,2,3,4,5,6,7,8}– if I select 5 different integers at random from this set, must two

of the numbers sum exactly to 9?– if I select 4 integers?

There exist two people in New York City who have the same number of hairs on their heads.

There exist two subsets of {1,…,10} with three elements which sum to the same value.

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Discrete StructuresCMSC 250Lecture 39

May 2, 2008

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Another (more useful) form of the pigeonhole principle

The generalized pigeonhole principle:– For any function f from a finite set X to a finite set Y and for any

positive integer k, if n(X) > k * n(Y), then there is some y Y such that y is the image of at least k+1 distinct elements of X.

Contrapositive form:– For any function f from a finite set X to a finite set Y and for any

positive integer k, if for each y Y, f–1(y) has at most k elements, then X has at most k n(y) elements.

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Examples

Using the generalized form:– assume 50 people in the room, how many must share the same birth

month?

– n(A)=5 n(B)=3 F: P (A) P (B)

how many elements of P (A) must map to a single element of P (B)?

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Composition of functions

f: X Y1 and g: Y Z where Y1 Y– g ○ f: X Z where (x X)[g(f(x)) = g ○ f(x)]

f(x) g(y)

g(f(x))

y

Y1

Y ZX

x z

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Composition on finite sets- example

ExampleX = {1,2,3}, Y1 = {a,b,c,d}, Y = {a,b,c,d,e}, Z = {x,y,z}

f(1) = c g(a) = y g○f(1) = g(f(1)) = z

f(2) = b g(b) = y g○f(2) = g(f(2)) = y

f(3) = a g(c) = z g○f(3) = g(f(3)) = y

g(d) = x

g(e) = x

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Composition for infinite sets- example

f: Z Z f(n) = n + 1

g: Z Z g(n) = n2

g ○ f(n) = g(f(n)) = g(n+1) = (n+1)2

f ○ g(n) = f(g(n)) = f(n2) = n2 + 1

Note: g ○ f f ○ g

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Identity function

iX the identity function for the domain X

iX : X X (xX) [iX(x) = x]

iY the identity function for the domain Y

iY : Y Y (yY) [iY(y) = y]

composition with the identity functions

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Composition with inverse

Recall: if f is a bijection then f1 exists.

Let f: X Y be a bijection.

What is f ○ f1?

What is f1 ○ f?

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One-to-one in composition

If f: X Y and g: Y Z are both one-to-one, then g ○ f: X Z is one-to-one.

If f: X Y and g: Y Z are both onto, then g ○ f: X Z is onto.

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Cardinality

Comparing the “sizes” of sets:– finite sets ( or there is a positive integer n such that there is a

bijective function from the set to {1,2,…,n})– infinite sets (there is no such n such that there is a bijective

function from the set to {1,2,…,n})

sets A,B, A and B have the same cardinality iff there is a one-to-one correspondence from A to B

In other words, Cardinality(A) = Cardinality(B) ( a function f ) [f: A B f is a bijection]

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Countable sets

A set S is called countably infinite iff Cardinalit(S) = Cardinality(Z+).

A set is called countable iff it is finite or countably infinite.

A set which is not countable is called uncountable.

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Discrete StructuresCMSC 250Lecture 40

May 5, 2008

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Countability of sets of integers and the rationals

N is this a countably infinite set? Z is this countably infinite set? Neven is this a countably infinite set? Card(Q+) =?= Card(Z)

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Real numbers

We’ll take just a part of this infinite set

Reals between 0 and 1 (noninclusive)X = {x R | 0 < x < 1}

All elements of X can be written as 0.a1a2a3… an…

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Cantor’s proof

Assume the set X = {x R | 0 < x < 1} is countable Then the elements in the set can be listed

0.a11a12a13a14…a1n…

0.a21a22a23a24…a2n…

0.a31a32a33a34…a3n…

… … … … Select the digits on the diagonal Build a number d, such that d differs in its nth position from

the nth number in the list

1if2

1if1

nn

nnn a

ad

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All reals

Cardinality({x R | 0 < x < 1}) = Cardinality(R)