Group homework#1 Due Due on Feb. 4, 2011 for Discrete Mathematics
Members:Rosela
Rowell, Carlos Rodriguez, Mark Salpeter, Chet Michals, Sarah Kidd
1.1 #10. Let p,q, and r be the propositions propositions
p: You get an A on the final exam. q: You do every exercise in this book boo k r: You get an A in this class Write these propositions using p,q A.
and r and logical connectives.
You get an A in this class, but you do not do every exercise in this book.
r r B. You get an A on the final, you do every exercise in this book, and you get an
A
in this class.
p q r C.
To
get an A in this class, it is necessary for you to get an A on the final.
r p D. You get an A on the final, but you dont do every exercise in this book; nevertheless, you get an A in this class.
(p q) r E.
Getting an A on the final and doing every exercise in this book is sufficient for getting an
A
in
this class.
( p F.
You will get an A in this class if and only if you either do every exercise in this book or you get an A on the final.
r 1.1 #16. For each of these sentences, determine whether an inclusive
or an exclusive or is
intended. Explain your answer. A.
Experience with C++ or Java is required. Inclusive (or).
You can have C++ or Java to get the job or both would be great and still
get the job. B.
Lunch
includes soup or salad.
Exclusive (or). You can pick only one choice. You cannot have both as freebie with your lunch otherwise, you pay an extra charge to have an additional item. C.
To
enter the country you need a passport or a voter regi stration card.
Inclusive(or). You
only need one requirement to enter the country, so either you have
the passport or a voter registration card or better more, both ar e acceptable. D.
Publish
or perish.
Exclusive(or). This is the same as dead or alive. You cannot be dead and alive at the same time. So, you need just one option.
1.1
#
20. Write each of these statements in the form if
p, then q in English. (hint: refer to the list
of common ways to express conditional statements provided in this section.) A.
I will remember to send you the address only if you send me an e-mail message.
Answer:
If you send me an e-mail message, then I will remember to send you the address.
B. If you keep your textbook, it will be a useful reference in your future courses. Answer:
If you keep your textbook, then it will be a useful reference in your future courses.
C.
you get the job implies that you had the best credentials.
That
Answer:
If you get the job, then you had the best credentials.
D. It is necessary to have a valid password to log on to the server. Answer:
If you log on to the server, then it is necessary to have a valid password.
1.1# 34. Construct a truth table for ((pq) r)s. p
q
r
s
p
(p
((p
T
T
T
T
T
T
T
T
T
T
F
T
T
F
T
T
T
T
T
F
T
F F
T
T
F F
T
F F F F
T
T
T
T
T
F
T
F
F F
T
T
T
F
F F F F
T
F
T
T
T
T
T
T
T
T
F
T
T
F
T
F F
T
T
T
F
T
F F
T
T
T
T
T
T
F
T
T
F
F F
T
T
T
F
T
F F
T T T
F F F F F F F F
T
F F F F
T
T
T
The following two
statements form the basis of the most important methods of theorem proving.
Use truth tables to prove that they are tautologies.
A. Resolution: ((P
p q r (p T T T F T T T F F T T F T T T T F F T T T F T T F T F T F F F F T T F F F F T F
B.
Modus ponens:
v q) ^ ( q v r)) (P v r) ( )
(p
p
((p
T
T
T
T
F
F
T
T
T
T
T
T
T
T
T
T
T
T
T
T
F
F F F
F
T
T
T
F
T
T T
(( P ^ (Pq))q
p q p p
((p
T
T
T
T
T
T
F F
F F F
T
F T F F
T T
T T
Show that Modus ponens is a
tautology without using a truth table.
Show each step and indicate
which logical equivalence you use.
A. Modus ponens: ( P
^ (Pq))q
[
De Morgans Laws
[
De Morgans Laws [ ( ) T
T
Modus ponens:
Show
( P ^ (Pq))q is a tautology
that (Pr) ^ (qr) and (p v q) r are logically equivalent.
p
q
r
p
q
(p (q
p
T
T
T
T
T
T
T
T
T
T
F
F
F
F
T
F
T
F F
T
T
T
T
T
T
F
F
T
F
T
F
T
T
T
T
T
T
T
T
F
T
F
F
T
F
F F
T
T
T
T
T
F
T
T
T
F F
T
F F F F
(p
T
