Practice Exercises
This is an open-book quiz. You may (and should) refer to the text as you solve these problems.
Don’t hurry! You’ll find worked-out answers in App. B. The solutions in the appendix may not
represent the only way a problem can be figured out. If you think you can solve a particular
problem in a quicker or better way than you see there, by all means try it!
- Imagine a dance at which there are 15 boys and 15 girls. Every boy writes his name on a
slip of paper, and then puts the slip into a jar. (No two boys have the same name.) The
girls pick slips out of the jar, one at a time and one for each girl, to determine who their
dance partner will be. If we think of this action as a mapping from the set of boys to the
set of girls, what type of mapping is this?
- Imagine another dance at which there are 10 boys and 15 girls. Every girl writes her
name on a slip of paper, and then puts the slip into a jar. (No two girls have the same
name.) The boys pick slips out of the jar, with every other boy taking two slips instead
of only one, so all the slips get taken. This determines how the dance partners will
be assigned. Five of the boys end up dancing with one girl, but the other five have to
contend with two girls! If we think of this action as a mapping from the set of boys to
the set of girls, what type of mapping is it?
- If we think of the mapping in Prob. 2 in reverse (that is, from the set of girls to the set
of boys) what type of mapping is it?
- Imagine that you and I were once members of Internet Network Alpha, which had
60,000 members. We were not totally honorable characters, you and I. We conspired
to send a mass e-mail message (also called spam) to every single one of the 175,000
members of Internet Network Beta. Let A be the set of all members of Alpha at that
time, and let B be the set of all members of Beta. Suppose that we had the latest
programs to defeat antispam software in other people’s computers, so we succeeded in
our dubious quest. Every single member of Beta got our message. What sort of mapping
did we carry out from set A to set B? What was the maximal domain? What was the
essential domain? What was the co-domain? What was the range? (Note: As a result of
our behavior, we were kicked out of Internet Network Alpha, a punishment which, we
realize today, was well deserved.)
- Consider the set Q of all rational numbers and the set R of all real numbers. Suppose we
devise a relation from Q to R that takes every integer q in Q and doubles it to get an even
integerr in R. The maximal domain is Q, and the essential domain is Z, the set of integers,
which is a proper subset of Q. The co-domain is R, and the range is Zeven, the set of all even
integers, which is a proper subset of R. What type of relation from Q to R have we devised?
- Suppose that we create a relation between the set Q of rational numbers and the set Z
of integers. To generate an integer z from a rational q, we find the fractional equivalent
ofq in lowest terms, and then chop off the denominator. If we let q be the independent
variable and z be the dependent variable, what type of relation is this? Is it injective? Is
it surjective? Is it bijective?
- Suppose that we create a relation between the set Z of integers and the set Q of
rationals. To generate a rational q from a nonzero integer z, we find the reciprocal of
Practice Exercises 221
