CSCE 235
Homework
Assignment 5
Assigned: February 16, 2009
Due: 12:30 p.m. February 25,
2009
(Homework 5 minutes late will not be
accepted)
1. (15 points) For each of the following sets,
determine whether there is an one-to-one function from S to T, whether
there is a function mapping S onto T, and whether there is an
one-to-one correspondence between S and T. If the answer is YES give an example; if the
answer is NO explain briefly.
(a) S = { 1, 2 , 3 } and T = { d, e, f }
(b) S = { a, b } and T = { 1, 2, 3, 4 }
(c) S = { 1, 2, 3, 4 } and T = { a, b }
2. (24 points) Determine whether each of these
functions is onto and one-to-one (where Z is the set of integers, and R is the set of real numbers).
Explain or show an example to support your answers. (Hint:
For (e)-(g), think carefully about the domain, target, and range
involved.)
(a)
f: Z → Z , 
(b)
f: Z → Z , 
(c)
f: R → R , 
(d)
f: R → R , 
(e)
f: Z → Z , 
(f)
f: R → R , 
(g)
f: R → Z ,
(h)
f: Z+ → R , 
3. (15 points) We define functions mapping R
to R (where R is the set of real numbers) as follows: 
, 
, 
. Find
(a) 
(b) 
(c) 
(d) 
(e) 
4. (16 points) For each of the following functions mapping R to R (where R is the set of real numbers), if it has an inverse, then find the inverse function. If it does not have an inverse, prove that it is not one-to-one correspondence. Make sure that your steps are correct.
(a) 
(b) 
(c) 
(d)
where
is a constant.
5. (50 points total) Let A denote the set
R\{ 0, 1 } (where R is the set of real
numbers). Let 
denote the identify
function on A and define the
functions f, g, h, s, r:
by
, 
, 
, 
, 
(a) (40
points) Show that 
and 
. Complete the table,
thereby showing that the composition of any two of the given functions is one
of the given five or the identity. (The
table you will construct in this exercise is the multiplication table for an
important mathematical object known as a group.
This particular group is the smallest one which is not commutative.) Show all steps clearly.
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(b) (10 points) Which of the given six functions have inverses? Find (and identify) any inverses which exist.
Problems
based on (Rosen 2003), (Ross and Wright 1988) and (Goodaire and Parmenter
2002).