Exercise 11.  Suppose  f  is a one-to-one mapping from  D  onto  T  and that  A  is a subset of  D.  

11 (a).  Show that  f  is one-to-one from  A  onto  B,  where  .

Solution 11 (a).

See text and/or instructor's solution manual.

Clearly,  f   is onto, because if  ,  then by definition of  B  there exists a point    such that  .  

Suppose that    for some values    in  A.

Then, because  A  is a subset of  D,    both belong to  D.  

But  f  is one-to-one on  D.

Hence,  .

Therefore,  f  is one-to-one.

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell