Math 2534
Solution for Homework 1 Fall 2012
Problem 1: Use Truth Tables to verify the following:
1)  (P  Q)  P   Q
p q
p
q pq
( p  q)
p q
T T
F
F
T
F
F
T F F
T
F
T
T
Continue to look at all possible truth values and then compare
the last two columns to see if the truth values are identical. This means that
you have a tautology and ( p  q )  p  q
2)
P  ( Q  R)  (P  Q) ( P R)
p q r q  r p  q p  r p  (q  r ) ( p  q )  ( p  r )
T T T
T
T
T
T
T
T T F
T
T
F
T
T
Continue to look at all possible truth values and then compare
the last two columns to see if the truth values are identical. This means that
you have a tautology and p  (q  r)  (p  q)  (p  r)
3)
P  ( Q  P)  P ( Solution process is same as above)
Problem 2: Given the following statements:
P: Oliver is Hokie
Q: Mary is a Whahoo
The statement P is true and the statement Q is false. Express each of the following
compound sentences in symbolic logic and use truth tables to determine the truth value.
1) Oliver is not a Hokie but Mary is a Whahoo.
ANS :
p  q  (T )  F  F  F  F
2) Either Oliver is a Hokie or Mary is not a Whahoo.
ANS : p q  T F  T  T  F
3) It is not true that Oliver is a Hokie or Mary is a Whahoo.
ANS :
(p  q)  (T  F )  T  F
4) Oliver is a Hokie and we have that Oliver is a Hokie or Mary is a Whahoo.
ANS : p  (p  q)  p  T (by absorption)
Problem 3: Determine if each of the following statements are Inclusive OR or
Exclusive OR.
1) At the Gator Restaurant in New Orleans you may have an free appetizer of Gator
bites or Calamari with each meal. (Exclusive OR)
2) It is required that you come to orientation or attend summer school before starting
the fall semester. (Inclusive OR)
3) With our budget for this summer we can build a swimming pool in the backyard
or we can go to the Bahamas. (Exclusive OR)
4) I will either have the car washed or I will wash it myself. (Exclusive OR)
Problem 4: A logic puzzle.
Instructions: You may solve this puzzle anyway you wish. Then you must write a
paragraph that gives a clear description of your reasoning. Do not use a chart or refer to a
chart in your final written explanation. Do your best. This is an exercise to help you
understand how difficult it can be to explain reasoning. This course will help you
develop the skills you need to be precise and clear.
The VT Baseball Batters:
The baseball team depends on four players to score most of their runs. The positions of
the four are the three outfielders (right fielder, center fielder, and left fielder) and the
catcher. From the statements that follow, determine the first name (Henry, Ken, Leo, or
Stan), surname (one is Dodson), position and batting average of each player. (Their
batting averages are .280, .295, .310, and .325)
One of the following statements is false:
1) Neither Leo nor the catcher has a batting average over .300.
2) Three who are neighbors are Clements, the right fielder, and the player
who bats .325.
3) The center fielder bats .295.
4) Stan’s batting average is 30 points higher than that of Ken, who does not live near
any of the other three.
5) Brooks and Henry, who is not Ashley, both bat over .300 and are in competition
to see which will score the most runs this season.
6) Henry, who is neither the right fielder not the left fielder, has a lower batting
average than the catcher.
Solution:
Statement 6 is false and Henry Dodson whose is the left fielder with average .325.
Ken Ashley is the catcher with average .280, Leo Clements is the center fielder and
average is .295, and Stan Brooks is the right fielder with average .310
Discussion: The reason the statement 6 is the false statement follows from the fact
that statement 1 claims that Leo and the Catcher both have averages below .300,
while statement 6 indicates that Henry has a lower batting average than the catcher.
This would mean that there are three people who have an average lower than .300
which is not possible. Statement 3 also indicates that the center fielder has an
average .295. This would mean that statement 1 and 3 would both be false if
statement 6 is true.
Since Leo and the catcher each have one of the only two scores below .300 as well as
the center fielder, Leo must be the center fielder with score of .295 and the catcher
has the only other possible score of .280.
Since Ken does not live near the other three. Therefore he does not have a score of
.325. Stan bats 30 point higher than Ken so the only possible score for Ken is 280
and he must be the catcher. Ken’s last name is not Clements since he does not live
next to other three. Brooks has a score over 300 so Ken is not Brooks and Henry is
not Ashely so that leaves Ken to have the last name of Ashely. Since Ken is the
only one not to live next to other three, Leo must be one of the three neighbors. Since
Leo is not the right fielder and does not have a batting average of .325, his last name
must be Clements. It is given that Stan’s batting average is 30 points higher than
Ken’s so it must be .310 and Henry will have .325. Since Stan is also one of three
neighbors, he must be the right fielder and therefore Henry is the left fielder. Stan
must also have the last name of Brooks. Since Brooks had a score above .300. That
leaves Dodson for Henry.
Summary:
Henry Dodson
Ken Ashely
Leo Clements
Stan Brooks
LF
Catcher
CF
RF
.325
.280
.295
.310