R. Johnsonbaugh_ Discrete Mathematics 5th edition_ 2001 by yurtgc548


p   A function f from X to Y (in
    symbols f : X ® Y) is a
    relation from X to Y such that
    Dom(f) = X and if two pairs
    (x,y) and (x,y’) Î f, then y = y’

    n   Example:
        Dom(f) = X = {a, b, c, d},
        Rng(f) = {1, 3, 5}
        f(a) = f(b) = 3, f(c) = 5, f(d) = 1.
       Domain and Range
n Domain of f = X
n Range of f =
    { y | y = f(x) for some x ÎX}
n A function f : X ® Y assigns to
  each x in Dom(f) = X a unique
  element y in Rng(f) Í Y.
n Therefore, no two pairs in f have
  the same first coordinate.
             Modulus operator
p Let x be a nonnegative integer and y a positive
p r = x mod y is the remainder when x is divided by
      1 = 13 mod 3
      6 = 234 mod 19
      4 = 2002 mod 111
p   mod is called the modulus operator
            One-to-one functions
p A function f : X ® Y is one-to-one Û
  for each y Î Y there exists at most one x Î X
  with f(x) = y.
p Alternative definition: f : X ® Y is one-to-one Û
  for each pair of distinct elements x1, x2 Î X there
  exist two distinct elements y1, y2 Î Y such that
  f(x1) = y1 and f(x2) = y2.
    n 1. The function f(x) = 2x from the set of real numbers to itself is
    n 2. The function f : R ® R defined by f(x) = x2 is not one-to-one,
      since for every real number x, f(x) = f(-x).
                Onto functions

A function f : X ® Y is onto Û
for each y Î Y there exists at least one x Î X
with f(x) = y, i.e. Rng(f) = Y.
 n   Example: The function f(x) = ex from the set of real
     numbers to itself is not onto Y = the set of all real
     numbers. However, if Y is restricted to Rng(f) = R +,
     the set of positive real numbers, then f(x) is onto.
               Bijective functions

A function f : X® Y is bijective Û
                     f is one-to-one and onto
n   Examples:
     p   1. A linear function f(x) = ax + b is a bijective function from
         the set of real numbers to itself
     p   2. The function f(x) = x3 is bijective from the set of real
         numbers to itself.
              Inverse function
p Given a function y = f(x), the inverse f -1 is the
  set {(y, x) | y = f(x)}.
p The inverse f -1 of f is not necessarily a
    n   Example: if f(x) = x2, then f -1 (4) = Ö4 = ± 2, not a
        unique value and therefore f is not a function.
p   However, if f is a bijective function, it can be
    shown that f -1 is a function.
             Exponential and
           logarithmic functions

p   Let f(x) = 2x and g(x) = log 2 x = lg x

    n   f ◦ g(x) = f(g(x)) = f(lg x) = 2 lg x = x
    n   g ◦ f(x) = g(f(x)) = g(2x) = lg 2x = x

p   Therefore, the exponential and logarithmic
    functions are inverses of each other.
    Composition of functions
p   Given two functions g : X ® Y and f : Y ® Z,
    the composition f ◦ g is defined as follows:
        f ◦ g (x) = f(g(x)) for every x Î X.
    q   Example: g(x) = x2 -1, f(x) = 3x + 5. Then
        f ◦ g(x) = f(g(x)) = f(3x + 5) = (3x + 5)2 - 1
q Composition of functions is associative:
         f ◦ (g ◦h) = (f ◦ g) ◦ h,
q But, in general, it is not commutative:
              f ◦ g ¹ g ◦ f.
              Binary operators

A binary operator on a set X is a function f that
associates a single element of X to every pair of
elements in X, i.e. f : X x X ® X and f(x1, x2) Î X
for every pair of elements x1, x2.

 n   Examples of binary operators are addition, subtraction
     and multiplication of real numbers, taking unions or
     intersections of sets, concatenation of two strings
     over a set X, etc.
                 Unary operators
p   A unary operator on a set X associates to
    each single element of X one element of X.
    n   Examples:
        1. Let X = U be a universal set and P(U) the power
        set of U. Define f : P(U) ® P(U) the function
        defined by f (A) = A', the set complement of A in U,
        for every A Í U. Then f defines a unary operator
        on P(U).
            String inverse
Let X be any set, X* the set of all strings over X.
 If a = x1x2…xn Î X*, let f(a) = a-1 = xnxn-
 1…x2x1, the string written in reverse order.
Then f :X* ® X* is a function that defines a unary
 operator on X*.

Observe that aa -1 = a -1a = l

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