Last modified Mar 20, 2008

If symbols do not show up properly in your browser or when you print this, see the pdf version at Functions and Inverse Functions, or see the printed version which will be handed out in class.

Asterisks denote definitions which are not in the textbook.

1. We review the definitions leading up to the textbook's definition of inverse function.
   1. *A relation is a set of ordered pairs.
      
      
   2. If there are sets  R,  A,  and  B  such that
      

      R ⊆ A × B,
      

      then  R  is said to be a relation between  A  and  B.
      (Clearly, if  R  is a relation between  A  and  B,  then  R  is a relation as defined in (a).)
      

   3. *The domain of a relation  R,  denoted by  dom(R),  is defined by
      

      a  is an element of  dom(R)  iff there exists  b  such that  (a, b)  is an element of  R.
      

   4. *A relation  R  is called a function, or is said to have the function property, iff
      

      For all  a,  b,  c,  and  d,  if  (a, b)  is an element of  R  and  (c, d)  is an element of  R  and  a = c,  then  b = d.
      

      In other words, a relation  R  is a function iff

      if  (a, b) ∈ R  and  (a, d) ∈ R  then  b = d.
      

   5. Given sets  A  and  B,  a relation  F  is said to be a function from  A  to  B  iff
      • F  is a relation between  A  and  B;
        
      • F  has the function property;
        
      • dom(F) = A.
      The notation  F : A  B  means that  F  is a function from  A  to  B,  although the exact verbal interpretation of the symbols  F : A  B  may differ slightly depending on the context.
      
      
   6. Theorem. *For a function  F : A  B,  define the relation  F ^-1  by
      (I)         (b, a) ∈ F ^-1  iff  (a, b) ∈ F,
      (i.e.,
              z ∈ F ^-1  iff there exist  a,  b  such that  z = (b, a)  and  (a, b) ∈ F)
      If  F  is bijective, then  F ^-1  is a function from  B  to  A.  (You should be able to prove this.)
      
      
   7. The definition of inverse function given in the textbook can be expressed in the following way:
      If a function  F : A  B  is bijective, then the function  F ^-1 : B  A  defined above is called the inverse function of  F : A  B.
      
      *To distinguish this concept from another inverse function concept discussed below, we will temporarily refer to this as the L-inverse function of  F : A B.
      (The letter "L" refers to the author of the textbook.)
      
      *In other words, a function  G : B A  is the L-inverse function of a bijective function  F : A B  iff  dom(G) = B  and  G  is the inverse relation  F ^-1  of the relation  F  as defined in (I) in part (f) above.
      
      
   8. *Expressed in standard function notation instead of with the use of ordered pairs, equation (I) above gives the following defining condition for the L-inverse:
      For every  a  in  A  and  b  in  B ,
      (I')         F ^-1(b) = a  iff  F(a) = b.
   
   
2. We will give another definition of "inverse function", and then we will comment on the definitions and the relationships between them.
   
   *Given a function  f : A  B,  a function  g : B  A  is called an inverse of the function  f : A  B  iff
   

   For all  x  in  A,  g(f(x)) = x,
   

   and
   

   For all  y  in  B,  f(g(y)) = y.
   

   We will temporarily refer to such a function  g : B  A  as an algebraic-inverse function of  f : A  B .
   
   It should be clear that this definition is equivalent to the following:
   For any set  S,  let  id_S  denote the identity function on  S.
   Given a function  f : A  B,  a function  g : B  A  is called an (algebraic-)inverse of the function  f : A  B  iff
   

   g o f = id_A
   

   and

   f o g = id_B.
   

Uniqueness. It is clear that the L-inverse of a function is unique, if it exists, since the L-inverse function is obtained in a particular way from the original function. On the other hand, if the inverse function is defined as an algebraic-inverse, then it has to be proved that if such an inverse function exists, then it is unique.

Equivalence of L-inverse and algebraic-inverse. You should be able to prove (this is related to Ex. 7.29 in the textbook) that the L-inverse idea and the algebraic-inverse idea are equivalent in the following sense:
A function  f : A  B  has an algebraic-inverse function iff  f : A  B  is bijective.
A function  g : B  A  is the L-inverse function of  f : A  B  iff  g : B  A  is an algebraic-inverse function of  f : A  B.

Uniqueness. Note that this implies that if an algebraic-inverse function exists, then it is unique.

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