1 cmsc 250 chapter 7, functions. 2 cmsc 250 function terminology l a relationship between elements...
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1CMSC 250
Chapter 7, Functions
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2CMSC 250
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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3CMSC 250
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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4CMSC 250
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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5CMSC 250
Discrete StructuresCMSC 250Lecture 38
April 30, 2008
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6CMSC 250
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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7CMSC 250
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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8CMSC 250
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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9CMSC 250
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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10CMSC 250
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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11CMSC 250
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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12CMSC 250
Discrete StructuresCMSC 250Lecture 39
May 2, 2008
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13CMSC 250
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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14CMSC 250
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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15CMSC 250
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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16CMSC 250
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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17CMSC 250
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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18CMSC 250
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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19CMSC 250
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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20CMSC 250
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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21CMSC 250
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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22CMSC 250
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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23CMSC 250
Discrete StructuresCMSC 250Lecture 40
May 5, 2008
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24CMSC 250
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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25CMSC 250
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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26CMSC 250
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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27CMSC 250
All reals
Cardinality({x R | 0 < x < 1}) = Cardinality(R)