Exercises for self study. Complete these exercises in the textbook and
compare your solutions against those in the back of the book. These
exercises will not be graded.
pp. 416–418, Exercises 3 and 33
p. 427, Exercise 15
pp. 435, 436, Exercise 27
Exercises to be graded. Complete these exercises and turn in your
solutions.
Since 2015, the Connecticut Department of Motor Vehicles has issued
vehicle license plates under the format of two letters followed by five
digits (e.g., AB $\!\cdot\!\!$ 12345). For simplicity, assume that all
two-letter five-digit combinations are acceptable. Answer the following
questions. Explain how you derived your answers.
(a) How many distinct plates can be issued?
(b) Of all possible plates, how many have all letters and digits distinct?
(c) Of all possible plates, how many have repeated letters (e.g.,
AA $\!\cdot\!$ 75261)?
(d) Of all possible plates, how many contain exactly two zeros (e.g.,
CT $\!\cdot\!$ 06106)?
A test has 100 true/false questions. Answer the following questions.
Explain how you derived your answers.
(a) How many ways can one answer the questions if no answer may be
left blank?
(b) How many ways can one answer the questions if answers may be left
blank?
(c) How many ways can one answer the questions with exactly 47 true answers
if no answer may be left blank?
(d) How many ways can one answer the questions with exactly 47 true answers
if some answers may be left blank?
Recall that two compound propositions are logically equivalent if
their truth values agree for all possible combinations of the truth values
of their propositional variables, i.e., if their truth tables are the same
(for example,
$p \leftrightarrow q \equiv (p \rightarrow q) \wedge (q \rightarrow p)$
since their truth values agree for all four combinations of the truth
values of $p$ and $q$). How many non-logically-equivalent compound
propositions of $n$ propositional variables are there? In other words, how
many different truth tables of compound propositions of $n$ propositional
variables are there? Explain how you derived your answer.
(a) Suppose that friendship is symmetric; that is, if a person $a$
is a friend of a person $b$, then $b$ is also a friend of $a$. We also
assume that no one is a friend of oneself. Prove that, in a group of two or
more people, at least two people must have the same number of friends in the
group.
(b) Prove that, in a group of 25 or more people, at least three people must
have their birthdays in the same month.
Plagiarism and academic dishonesty.
Remember, under any circumstance, you must not copy part or all of another's
work and present it as your own. For more details, read
our course policy on plagiarism and
academic dishonesty.
