Test 1 __________________________________________________ Sept. 22, 2011
1. Mark the following statements true or false. If false, correct the statement so that
it is true.
a. The conditional statement p  q is only true when p is true.
False, the conditional statement p  q is false only when p is true and q is
false.
b. The precedence of logical operators is , , , and .
True. (page 11, table 8)
c. The symbol used for disjunctions is .
False, The symbol used for disjunctions is .
d. The predicate refers to a property that a proposition can have.
True. (page 37, next to the last paragraph.)
e. Zero and one are used for Boolean variables in circuits.
True. (page 11)
f. A compound proposition that is always false, no matter what the truth values of the
propositional variables that occur in it, is call a contradiction.
True. (page 25)
g. The negation of x is "there is an x".
True.
(page 47)
h. The quantifiers  and  have higher precedence than all logical operators from
propositional calculus.
True. (page 44)
i. An example of a vacuous proof is that 'p  q' is true based upon the fact that p is false.
True.
(page 110)
j. A propositional function is a statement containing one or more variables that becomes
a proposition when each of its variables is assigned a value and bound by a quantifier.
False. A propositional function is a statement containing one or more
variables that becomes a proposition when each of its variables is assigned a
value or bound by a quantifier.
2. Using P(x), Q(x), and R(x), express the following argument assuming the domain of x
to be all creatures.
"All lions are fierce."
"Some lions do not drink coffee."
"Some fierce creatures do not drink coffee."
x(P(x)  Q(x))
x (P(x)   R(x))
x (Q(x)   R(x))
3. Show that (p  q)  (p  q) is a tautology.
(p  q)  (p  q)  (p  q)  (p  q)
Logical equivalence
 ( p   q)  (p  q) De Morgan's law
 ( p  p)  ( q  q) Associative and commutative laws
TT
Negation laws
T
Domination ( or idempotent) laws
or
p
T
T
F
F
q
T
F
T
F
p  q)
T
F
F
F
(p  q)  (p  q)
T
T
T
T
4. The following statement relates to inhabitants of an island on which there are
three kinds of people: knights who always tell the truth, knaves who always lie, and
spies who can either lie or tell the truth. You encounter three people, A, B, and C.
You know one of these people is a knight, one is a knave, and one is a spy. Each of the
three people knows the type of person each of other two is. For each of these
situations, if possible, determine whether there is a unique solution and determine
who the knave, knight, and spy are. When there is no unique solution, list all
possibilities and when there is no solution possible, say so. Give the reason for your
answer.
A says, "C is the knave", B says, "A is the Knight", and C says, "I am a Spy".
A
B
C
knight
knight
knave
knave
spy
spy
knave
spy
knight
spy
knight
knave
spy
knave
spy
knight
knave
knight
Truth values of
statements
A(F), B(T), C(T)
A(T), B(T), C(F)
A(F), B(F), C(T)
A(F), B(F), C(F)
A(T), B(F), C(F)
A(F), B(F), C(F)
Is it possible?
no
no
no
no
no
5. Suppose there are signs on the doors to two rooms. The sign on the first door reads “ In
this room there is a lady, and in the other one there is a tiger”; and the sign on the second
door reads “ In one of these rooms, there is a lady, and in one of them there is a tiger.”
Suppose that you know that one of these signs is true and the other is false. Behind which
door is the lady? Show your reasoning.
if A is:
then B is;
lady  ¬ tiger
tiger  lady
tiger  ¬ lady
lady  tiger
Truth values of
statements
A(T)  B(F)
A(T)  B(T)
Is assertion
possible?
y
y
Since “ In one of these rooms, there is a lady, and in one of them there is a tiger.” is
true when “ In this room there is a lady, and in the other one there is a tiger” is true or
false and we know that only one sign is true, the other sign must be false and the lady
is in the second room.
6. Let p ,q, and r be the propositions
p : You have the flu.
q : You miss the final examination.
r : You pass the course.
Express each of these propositions as an English sentence.
a. (p  ¬ r)  (q  ¬ r)
If you have the flu or you miss the final examination, then you will not pass the
course.
b. (p  q)  (¬ q  r)
You have the flu and you miss the final examination or you don't miss the final
examination and you pass the course.
7. Let p ,q, and r be the propositions
p : You get an A on the final exam..
q : You do every exercise in this book.
r : You get an A in this class.
Write these propositions using p ,q, and r and logical connectives (including
negations).
a. Getting an A on the final and doing every exercise in this book is sufficient for
getting an A in this class..
(p  q)  r
b. To get an A in this class, it is necessary for you to get an A on the final.
pr
('q is necessary for p' form)
8. Using the domain 'a student', translate the following into a logic expression. Label
each statement before using:
A student in your school knows Java and knows C++ but does not know Mathlab.
S(x) a student in your school.
J(x) a student that knows Java.
C(x) a student that knows C++.
M(x) a student that knows Mathlab.
x (S(x)  P(x)  J(x)   M(x))