Chapter 7, Functions

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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 diagrams
• Determining 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

• FX ? 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

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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

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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 F?1

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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.

12
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) gt k n(Y), then there is some y ? Y such
  that y is the image of at least k1 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, f1(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)

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Composition on finite sets- example

• Example
• X 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(n1) (n1)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 (?x?X) iX(x) x
• iY the identity function for the domain Y
• iY Y ? Y (?y?Y) iY(y) y
• composition with the identity functions

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Composition with inverse

• Recall if f is a bijection then f?1 exists.
• Let f X ? Y be a bijection.
• What is f ? f?1?
• What is f?1 ? 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.

23
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

• Well take just a part of this infinite set
• Reals between 0 and 1 (noninclusive)
• X x ? R 0 lt x lt 1
• All elements of X can be written as
• 0.a1a2a3 an

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Cantors proof

• Assume the set X x ? R 0 lt x lt 1 is
  countable
• Then the elements in the set can be listed
• 0.a11a12a13a14a1n
• 0.a21a22a23a24a2n
• 0.a31a32a33a34a3n
• 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

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All reals

• Cardinality(x ? R 0 lt x lt 1) Cardinality(R)