Lecture 14 Functions

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Lecture 14 Functions

• July 1st, 2003

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What is a function?

• Functions are a special case of binary relations
  from a set S to a set T.
• The idea of associating an element from one set
  with an element (or elements) from another is a
  fundamental one in mathematics.

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Function-Definition

• A function from A to B is a mapping from one set
  (the domain of the function) to another (the
  codomain of the function), where each element of
  the domain is mapped to one element of the range.

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Function-Definition

• The critical idea is that a function maps each
  element in its domain to a unique element in its
  range.
• Example Exercise 6( Page 313)
• There are three parts to a function
• A set of starting values (S)
• A set from which associated values come (T)
• The association itself.

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Function-Definition

• The association is a set of ordered pairs, each
  of the form (s, t) where s ? S, t ? T, ant t is
  the value from T that the function associates
  with the value s from S.
• f S ? T
• t f(s), t is the image of s under f, s is a
  preimage of t under f
• The association is a subset of S ? T.

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Function-- Reminder

• A function is a special kind of binary relation.
• A binary relation that is one-to-many (or
  many-to-many) cannot be a function.
• We can have functions with more than one
  variable, for example
• f S1 ? S2 ? ? Sn ? T
• Associates with each ordered n-tuple of elements
  (s1,s2,,sn)
• si ? Si, a unique element of T.

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Function-- Examples

• From R to Z
• Floor function
• Ceiling function
• From Z to Z
• Modulo function

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Function- Equal

• Two functions are equal if they have the same
  domain, the same codomain, and the same
  association of values of the codomain with values
  of the domain.
• So, f S ? T means
• ?s ? S ?1 t ? T such that f(s) t

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Onto functions

• Every member of S has an image under f, and all
  the images are members of T the set R of all
  such images is called the range of the function
  f.
• Thus, R f(s) s ? S, R ? T
• If it should happen that R T, then the function
  is called an onto function.

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Onto functions (cont.)

• Definition
• A function f S ? T is an onto, or surjective,
  function if the range of f equals the codomain of
  f.
• ?t ? T ? s ? S such that f(s) t
• Example
• A a, b, c, d B x, y, z
• f A?B is defined as
• f (a) x f (b) y f (c) z f (d) z
• so f is onto

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Onto Function--more examples

• f Z?Z, f(x) x2, f is not onto because theres
  no negative numbers on the range of f.
• f Q?Q, f(x) 3x 2. f is onto
• f Z?Q, f(x) 3x 2. f is not onto

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One to One Function

• Definition
• A function f S ? T is one-to-one, or injective,
  if no member of T is the image under f of two
  distinct elements of S.
• Different objects in the domain have different
  images on the codomain.
• a ? a ? f(a) ? f(a) or f(a) f(a) ? a a
• Example
• 1. f Z ? Z, f(x) x2, is not injective f(2)
  f(-2) 22 4
• 2. f Z ? Z, f(x) x3, is injective a ? b ? a3
  ? b3

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Bijective Function

• Definition
• A function f S ? T is bijective (a bijection) if
  it is both one-to-one and onto.
• 1. ( f(a) f(a)) ? (a a)
• 2. y ? B ? x ? A ? f(x) y

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Function More example
Not a function
Not a function
Function, not one-to-one, not onto
Function, not one-to-one, onto
Function, one-to-one, not onto
Function, one-to-one, onto
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Composition of Functions

• Definition
• Let f S?T and g T?U. Then the composition
  function g ? f, is a function from S to U defined
  by
• (g ? f) (s) g( f(s) )

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Composition of Functions (cont.)

• It is not always possible to take two arbitrary
  functions and compose them the domain and ranges
  have to be compatible.
• Function composition preserves the properties of
  being onto and being one-to-one.
• The composition of two bijections is a bijection.

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Composition--Examples

• f Z?Z, f (x) 3x-1
• g Z?Z, g (x) 2x2
• What is the value of (gof)(4)
• b. What is the value of (fog)(4)

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Inverse Functions

• Sa, b T7,4
• Lets consider the function f A?B
• Such that f(a)4 and f(b)4, so f(a,4),(b,4)
• Lets consider (f-1,B,A) such that f(4)a and
  f(4) b we can see that f-1 is not a function
  since an element of the domain has two different
  images.
• The necessary and sufficient condition for a
  function to have an inverse is that the function
  must be a bijection.

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Inverse function--Identity

• Let f S?T be a bijection. Every t ? T has a
  unique preimage s ? S, so each member of the
  codomain has an association with a unique member
  of the domain.
• This associations induce a function g T?S
• We can define a function that maps each element
  of a set S to itself, so it leaves the element of
  S unchanged. This function is denoted by is. (g ?
  f is)
• The same thing happens with T, f ? g ig

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Inverse function-- Definition

• Definition
• Let f be a function, f S?T. If there exists a
  function g T?S such that g ? f is and f ? g
  iT. Then g is called the inverse function of f,
  denoted by f-1.
• Theorem
• Let f S?T. Then f is a bijection if and only
  if f-1 exists.
• f(x) y , f-1(y) x

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Inverse function-- Example

• The inverse of a function f is another function,
  denoted by f-1, such that f-1(f(x)) x for all x
  in the domain of f.
• Examples
• f R?R given by f(x)3x4 is a bijection. What
  is f -1

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Permutation Functions

• Definition (Permutation of a Set)
• For a given set A, SA f f A?A and f is a
  biyection
• SA is thus the set all bijections of set A into
  (and therefore onto) itself such functions are
  called permutations of A.
• Function composition is a binary operation on
  the set SA.
• Permutation functions represent ordered
  arrangements of the
• objects in the domain.

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Permutation Functions(cont.)

• Example
• Where the domain is A 1, 2, 3, 4. The
  function f could be written as f (1,2),
  (2,3), (3,1), (4,4).
• If f and g are members of SA for some set A, then
  g ? f ? SA, and the action of g ? f on any member
  of A is determined by applying function f and
  then function g.

Cycle notation
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Permutation Functions(cont.)

• If f and g are members of SA, and f and g are
  disjoint cycles then
• f ? g g ? f.
• Examples
• A1,2,3,4,5
• f (5,2,3) g (3,4,1)
• f (1,3) g (2,5)

The cycles have no elements in common
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Permutation Functions(cont.)

• The permutation that maps each element of A to
  itself is the identity function on A, iA, also
  called the identity permutation.
• A 1,2,3,4,5, f ? SA given by f (1,2), f ?
  f ?
• Every permutation on a finite set that is not the
  identity permutation can be written as
  composition of one or more disjoint cycles.

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Order of magnitude

• Definition (Order of Magnitude)
• Let f and g be functions mapping nonnegative
  reals into nonnegative reals. Then f is the same
  order of magnitude as g, written f O(g), if
  there exist positive constants n0, c1, c2 such
  that for x ? n0, c1g(x) ? f(x) ? c2g(x)

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Exercise

• Exercise 4.2
• 6, 28,32,71