Section 7.3 continued

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Section 7.3 continued

• One-to-one Correspondences, Inverse Functions

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One-to-one Correspondences

• A one-to-one correspondence (or bijection) from a
  set X to a set Y is a function F X ? Y that is
  both one-to-one and onto.

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Example

• Let ? x, y and remember that ? is the set of
  all finite strings over ?. Define a function
• g ? ? ? by the rule
• g(s) the string obtained by writing the
  characters of s in reverse order
• Is g a bijection from ? to itself?
• YES!

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Proof for example

• To show that g is a bijection, we must show that
  it is both one-to-one and onto.
• First we will show that g is one-to-one. Suppose
  that for some strings s1 and s2 in ?, g(s1)
  g(s2). This means that the string obtained by
  writing the characters of s1 in reverse order is
  the same as the string obtained by writing the
  characters of s2 in reverse order. But if s1 and
  s2 are equal when written in reverse order, then
  they must be equal to start with. In other
  words, s1 s2.

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More of the proof

• To show that g is onto, suppose a string t is in
  ?. Let s g(t). By the definition of g, s is
  the string in ? obtained by writing the
  characters of t in reverse order. But when the
  order of the characters of a string is reversed
  once and then reversed again, we have the
  original string. Therefore, g(s) g(g(t)),
  which is the string obtained by reversing the
  order of the characters of t twice, so g(s)
  t. ?

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Inverse Functions

• Suppose F X ? Y is a one-to-one correspondence.
  Then there is a function
• F -1 Y ? X
• that is defined as follows
• Given any element y in Y,
• F -1(y) the unique element x ? X such
  that F(x) y.
• The function F -1 is called the inverse function
  of F.

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Finding an Inverse Function for a Function Given
By an Arrow Diagram

• Reverse the direction of the arrows.

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Finding an Inverse Function for a Function Given
in Words

• Try to figure out what will get you from the
  output to the original.
• Example For our earlier function g(s) the
  string obtained by writing the characters of s in
  reverse order, we want to reverse the order
  again. So
• g -1 (t) the string obtained by writing the
  characters of t in reverse order

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Finding an Inverse for a Function Given by a
Formula

• Try to undo what was done to get the function.
• Recall f R ? R defined by f (x) 5x 4.
• To find f -1, set y f (x) and solve for x.
• y 5x 4
• y 4 5x
• (y 4)/5 x.
• Therefore, f -1 (y) (y 4)/5.

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Theorem If X and Y are sets and F X ? Y is
one-to-one and onto, then F -1 Y ? X is
one-to-one and onto.

• Proof First we will show that F -1 is
  one-to-one. Suppose y1 and y2 are elements of Y
  such that
• F -1(y1) F -1(y2). Let x F -1(y1) F
  -1(y2). Then x ? X and by the definition of F
  -1,
• F(x) y1 since x F -1(y1) and
• F(x) y2 since x F -1(y2).
• Thus, y1 y2 since otherwise F would not be a
  function.

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Proof, part 2

• Next we will show that F -1 is onto. Suppose x
  is an element of X. Let y F(x). Then y is an
  element of Y and by definition of F -1, F -1(y)
  x. Since we have shown some element of Y maps to
  x, we are done. ?

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Mondays class

• Section 7.4 The Pigeonhole Principle