Discrete Structures Chapter 5

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Discrete StructuresChapter 5 Relations and
Functions
Nurul Amelina Nasharuddin Multimedia Department
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Objectives

• On completion of this chapter, student should be
  able to
• Define a relation and function
• Determine the type of function (one-to-one, onto,
  one-to-one correspondence)
• Find a composite function
• Find an inverse function

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Outline

• Cartesian products and relations
• Functions Plain, one-to-one, onto
• Function composition and inverse functions
• Functions for computer science
• Properties of relations
• Computer recognition Zero-one matrices and
  directed graphs
• Use in database example

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Identity Function (pg 394)

• The function 1A A ? A, defined by 1A (a) a for
  all a ? A, is called the identity function for A
• It is a function that always returns the same
  value that was used as its argument. In terms of
  equations, the function is given by f(x)  x
• 1X(x) x for all x in X

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Function Equality

• When are f and g equal?
• If f, g A ? B, we say that f and g are equal and
  write f g, if f (a) g (a) for all a ? A
• Eg Define f R ? R and g R ? R by the
  following formulas
• f(x) x for all x ? R
• g(x) vx2 for all x ? R
• Does f g? Yes, x vx2 for all x ? R

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Composite Function

• Function composition is an operation for
  combining two functions
• If f A ? B and g B ? C, we define the composite
  function, which is denoted g o f A ? C,
• by (g o f )(a) g (f(a)), for each a ? A

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Example (1)
Let A 1, 2, 3, 4, B a, b, c, and C w,
x, y, z with f A ? B and g B ? C given by f
(1, a), (2, a), (3, b), (4, c) and g (a, x),
(b, y), (c, z). For each element of A we
find (g o f) (1) g (f (1)) g (a) x (g o f)
(2) g (f (2)) g (a) x (g o f) (3) g (f
(3)) g (b) y (g o f) (4) g (f (4)) g (c)
z So g o f (1, x), (2, x), (3, y), (4, z)
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Example (2)

• Are f o g and g o f equal?
• Let f R ? R, g R ? R be defined by f (x) x2,
• g(x) x 5. Then
• (g o f)(x) g (f(x)) g(x2) x2 5, whereas
• (f o g)(x) f (g(x)) f(x 5) (x 5)2 x2
  10x 25
• Here g o f R ? R and f o g R ? R, but
• (g o f)(1) 6 ? 36 (f o g)(1), so even though
  both composites f o g and g o f can be formed, we
  do not have f o g g o f

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Example (3)

• Let f, g, h R ? R, where f(x) x2, g(x) x 5
  and h(x)
• Then, ((h o g) o f)(x) (h o g)(f(x)) (h o
  g)(x2)
• h(g(x2)) h(x2 5)

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Example (4)

• Let f, g, h R ? R, where f(x) x2, g(x) x 5
  and h(x)
• Find (h o (g o f))(x).
• Is (h o (g o f)) ((h o g) o f)?
• Yes. Using the definition of the composite
  function, we find that
• ((h o g) o f) (x) (h o g) (f (x)) h ( g
  (f (x))), whereas
• (h o (g o f )) (x) h ((g o f ) (x)) h (g
  (f (x))).

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Associative Property
Since ((h o g) o f)(x) h (g(f(x))) (h o (g o
f))(x), for each x in A, it now follows that (h o
g) o f h o (g o f)
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Composition with an Identity Function

• If f is a function from a set A to a set B, and
  1A is the identity function on A and 1B is the
  identity function on B, then
• f o 1A f 1B o f

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Example (1)

• Let X a,b,c,d and Y u,v,w and f (a,
  u), (b, v), (c, v), (d, u). Find f o 1X and 1Y o
  f
• (f o 1x)(a) f(1x(a)) f(a) u
• (f o 1x)(b) f(1x(b)) f(b) v
• (f o 1x)(c) f(1x(c)) f(c) v
• (f o 1x)(d) f(1x(d)) f(d) u
• So, (f o 1x)(x) f(x)
• Find (1Y o f ) and show that for (1Y o f)(x)
  f(x)

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Properties of Composite Function
Let f A ? B and g B ? C a) If f and g are
one-to-one, then g o f is one-to-one b) If f and
g are onto, then g o f is onto
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Converse of a Function

• Since a function is also a relation, first let us
  see the converse of a relation
• For sets A, B, if R is a relation from A to B,
  then the converse of R, denoted Rc, is the
  relation from B to A defined by R (b, a)(a,
  b) ? R
• Eg A 1, 2, 3, 4, B w, x, y, and R (1,
  w), (2, w), (3, x) then Rc (w, 1), (w, 2),
  (x, 3), a relation from B to A

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Example (1)

• For A 1, 2, 3 and B w, x, y, let f A ? B
  be given by f (1, w), (2, x), (3, y)
• Then f c (w, 1), (x, 2), (y, 3) is a
  function from B to A, and we observe that
• f c o f 1A and f o f c 1B
• f c o f(1) f c(w) 1
• f c o f(2) f c(x) 2
• f c o f(3) f c(y) 3
• f c o f (1,1), (2,2), (3,3) 1A

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Invertibility of a Function

• If f A ? B, then f is said to be invertible if
  there is a function g B ? A such that
• g o f 1A and f o g 1B
• A function f A ? B is invertible iff it is
  one-to-one and onto

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Example (1)

• Let f, g R ? R be defined by
• f(x) 2x 5,
• g(x) (1/2)(x 5). Then
• (g o f)(x) g(f(x)) g(2x 5)
• (1/2)(2x 5) 5 x, and
• (f o g )(x) f(g (x)) f((1/2)(x 5))
• 2(1/2)(x 5) 5 x
• So f o g 1R and g o f 1R. Consequently, f and
  g are both invertible functions

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Inverse Functions
Suppose f X ? Y is a one-to-one correspondence.
Then inverse function, f-1 Y ? X that is defined
as follows Given any element in Y, f-1(y) that
unique element x in X such that f(x) equals y In
other words, f-1(y) x ? y f(x)
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Finding an Inverse Function

• Find the inverse function for f(x) 4x 1for
  all real numbers x
• By definition of f-1(y) that unique real number
  x such that f(x) y
• f(x) y
• 4x - 1 y
• x (y 1)/4
• f -1(y) x, hence f -1(y) (y 1)/4
• Rename y by x, f -1(x) (x 1)/4

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Quiz 4A

1. Let f Z?Z be the successor function and let g
   Z?Z be the squaring function. Then f(n) n 1
   for all n ? Z and g(n) n2 for all n ? Z. (a)
   Find the compositions g o f and f o g. (b) Is g o
   f f o g? Explain.
2. Let X a,b,c,d and Y u,v,w, and suppose f
   X ? Y is given by (a, u), (b, v), (c, v), (d,
   u). Find f o 1X and 1Y o f.
3. f R ? R is defined by f(x) 3x 5. Find its
   inverse function.

Send your answers in the next class!