Solutions for Math 2534 Homework (sec 2.1)
Problem 1:
Let p, q and r be the propositions given below:
p: Rain is predicted for the day.
q: There are bikers on the road.
r: There will rest stops along the way.
Part A
Put the sentences below into symbolic logic notation.
a) There are rest stops along the way and rain is not predicted today.
b) Bikers are on the road, but rain is predicted for the day.
c) There are not bikers on the road or rain is not predicted for the day.
d) Neither bikers are on the road nor is rain predicted for the day.
e) Either rain is predicted for the day or bikers are on the road.
r p
q p
q p
q p
pq
Problem 2: For each of the sentences below, determine whether an ‘exclusive or’ or an
‘inclusive or’ is intended.
1) To apply for this job you need 3 years experience or a masters degree. (Inclusive OR)
2) Dinner includes a soup or salad in the price. (Exclusive OR)
3) The game and the concert are at the same time, you may go to one or the other.
(Exclusive OR)
Problem 3: Let p and q be the following propositions:
p: I never miss a game.
q: I go to VT.
Express the following in short efficient English sentences.
(use properties proved in class)
1) p
q
I never miss a game and I do not go to VT
2)
( p  q)  (p
3)
q  (p
q) =
q) by DeMorgan's Law : I never miss a game and I do not go to VT
q by Absorption Law: I do not go to VT
Problem 4:
Construct a truth table to verify the following equivalence relation:
( p  q)  r  [( p q) r ]
p
q
r
p  q ( p  q)  r
p
q ( p
q )
r
[( p 
q )
p
q
r
T
T
T
F
F
F
T
T
F
F
T
T
T
F
F
F
T
T
F
F
T
F
T
F
F
T
F
T
.
.
.
.
.
.
The truth values and work is incomplete. However, notice the number of columns
needed to have a precise truth table. When you have finished all possible truth value
combinations (there should be 8 rows). You will then write a sentence stating that column
8 and column 11 have identical truth values which is a tautology and therefore the
equivalence relation is true.
Problem 5
Logic Puzzle:
Solve the following puzzle and then write up a paragraph explaining your reasoning.
Use your own judgment on how to write this up. It needs to be clear to anyone who tries
to follow your write up.
A Fishing Vacation:
Carl and his three friends who are avid fishermen decided to visit four of the world’s
finest fishing locations over a period of four years. Each friend had a different favorite
location, so they chose the order in which they would take the trips by drawing the place
names from a hat. Based on the statements below, what is the first name (Andy, Bill,
Carl, or Dennis) and surname (one is Cole) of each friend, what was the favorite fishing
location of each, and what was the order in which the trips were to be taken?
One of the following statements below is false.
1) Andy and Barrott work for the same company and Bill is self-employed.
2) The Iceland trip was planned for the year before the Alaska trip.
3) The Alaska trip was not Andy’s favorite destination.
4) Carl and Barrott and their wives frequently play bridge together.
5) Whelan’s favorite fishing location was the third trip, which was neither Patagonia
nor New Zealand.
r]
6) Whelan and Cole are married; the other two are not.
7) Dennis was disappointed that his favorite destination, New Zealand, was not to be
the first trip, Crowley’s favorite destination was planned for the last trip, so he
was unsympathetic.
8) Barrott’s favorite destination was the first.
Solution: It is given that Iceland is scheduled before Alaska. So the position choices for
those two trips will be 1st & 2nd, 2nd&3rd, 3rd & 4th. It is given that neither the Patagonia
or New Zealand trip is the 3rd trip and that Dennis’s favorite trip, New Zealand, is not
first. Therefore we know that Patagonia must be first. The only choice for New Zealand
is now 2nd or 4th since Alaska must follow Iceland. We know that Dennis’s trip is not
first and Crowley’s trip which is not New Zealand is last. So New Zealand must be
second and Crowley’s favorite trip is Alaska which must be last.
We know that Crowley’s first name is not Andy since Andy’s favorite trip is not Alaska.
We know that Whelan’s favorite trip is the 3rd trip to Iceland. We know that Dennis’s
favorite trip is the 2nd one to New Zealand and that Barrott’s favorite was the 1st one to
Patagonia. So Dennis’s last name must be Cole. Barrott’s first name is Carl since it can’t
be Bill or Andy (This would make statement 4 the false statement) . Crowley’s first
name is not Andy, so it must be Bill and Whelan first name is Andy.
1) Carl Barrott Patagonia
2) Dennis Cole New Zealand
3) Andy Whelan Iceland
4) Bill Crowley Alaska