Problem 1.
Let p and q be the propositions:
p: You drive over 65 miles per hour
q: You get a speeding ticket
Write these propositions using p and q and logical connectives.
a) You do not drive over 65 miles per hour \ Ø p \
b) You drive over 65 miles per hour but you do not get a speeding ticket \ p Ù Øq \
c) You will get a speeding ticket if you drive over 65 miles per hour \ p ® q \
d) If you do not drive over 65 miles per hour, then you will not get a speeding ticket \ Øp ® Øq \
e) Driving 65 miles per hour is sufficient for getting speeding ticket \ p ® q \
f) You get a speeding ticket, but you do not drive over 65 miles per hour \q Ù Øp \
g) Whenever you get a speeding ticket, you are driving over 65 miles per hour
\ q ® p \
Problem 2.
Let p,q, and r be the propositions:
p: Grizzly bears have been seen in the area
q: Hiking is safe on the trail
r: Berries are ripe along the trail
Write these propositions using p,q, and r and logical connectives.
a) Berries are ripe along the trail, but grizzly bears have not been seen in the area
r Ù Øp
b) Grizzly bears have not been seen in the area and hiking on the trail is safe, but berries are ripe along the trail
Øp Ù q Ù r
c) If berries are ripe along the trail, hiking is safe if and only if grizzly bears have not been seen in the area
r ® (q « Ø p)
d) It is not safe to hike on the trail, but grizzly bears have not been seen in the area and the berries along the trail are ripe
Øq Ù Øp Ù r
e) For hiking on the trail to be safe, it is necessary but not sufficient that berries not be ripe along the trail and for grizzly bears not to have been seen in the area
(q ® (Ør Ù Øp)) Ù Ø((Ør Ù Øp) ® q)
f) Hiking is not safe on the trail whenever grizzly bears have been seen in the area and berries are ripe along the trail
(p Ù r) ® Øq
Problem 3.
Construct a truth table for each of these compound propositions.
a) p ® (Øq Ú r); Øp ® (q ® r); (p ® q) Ú (Øp ® r); (p ® q) Ù (Øp ® r);
b) (p « q) Ù (Øq « r); (Øp « Øq) « (q « r);
Problem 4.
Show that (p Ú q) Ù (Øp Ú r) ® (q Ú r) is a tautology
T F
q – F; r – F; p Ú q – T; Øp Ú r – T; p – T; r - T
Problem 5.
Show that (p Ù q) ® r and (p ® r) Ù (q ® r) are not equivalent
1 0 0 1 0 0 0
1 0
Problem 6.
Use De Morgan’s lows to find the negation of each of the following statements.
1) Jan is rich and happy
2) Carlos will bicycle or run tomorrow
3) Mei walks or takes the buss to a class
A – Jan is rich ; B- Jan is happy; A Ù B; Ø[ A Ù B]= ØA Ú ØB
Problem 7.
Show that Ø and Ú form a functionally complete collection of logical operators
A Ù B = Ø[ØA] Ù Ø[ØB] = Ø[ØA Ú ØB]
A ® B = ØA Ú B
Definition.
A compound statement is satisfiable if there is an assignment of truth values to the variables in the compound proposition that makes it true.
Problem 8.
It the compound proposition (p Ú q Ú Ør) Ù (p Ú Øq Ú Ør) Ù (Øp Ú Øq Ú r)
satisfiable? ----------------------------------------------------------
p=1 -> Øq=1 or r =1 0 0 0
q=1 -> p= 1 or r=0 0 0 1 0 1 1 1 1 0
r=0 -> Øp=1 or Øq=1
Problem 9.
The police have three suspects for the murder of Mr. Cooper: Mr. Smith, Mr. Jones, and Mr. Williams. Smith, Jones, and Williams declare that they did not kill Cooper. Smith also states that Cooper was a friend of Jones and that Williams disliked him. Jones also states that he did not know Cooper and that he was out of town the day Cooper was killed. Williams also states that he saw both Smith and Jones with Cooper the day of the killing and that either Smith or Jones must have killed him. Can you determine who the murderer was if
a) one of the three men is guilty, the two innocent men are telling the truth, but the statements of the guilty man may or may not be true
b) innocent men do not lie?
Solution:
S- Smith is telling the truth; J – Jones is telling the truth; W- Williams is telling the truth; a – Cooper friend of Jones; b – Williams did not like Jones; c – Jones did not know Cooper; d – Jones was out of town the day Cooper was killed; e – Williams saw Jones with Cooper the day of the killing; f - Williams saw Smith with Cooper the day of the killing; g – Smith killed Cooper; h – Jones killed Cooper.
Statements describing crime:
I. S ® a Ù b Ù Øh
II. J ® c Ù d Ù Øh
III. W ® e Ù f Ù (g Ú h)
a ® Øc
d ® Øh
c ® Øe equivalent to e ® Øc
h ® Øa equivalent to a ® Øh
From statements I,II,III we know that if W is telling the truth and h is true than both S and J are not saying the true which is not possible. So, statement III is of the form W ® e Ù f Ù g.
Now, our three statements can be extended to:
IV. S ® a Ù b Ù Øh Ù Øc
V. J ® c Ù d Ù Øh
VI. W ® e Ù f Ù g Ù Øc
So either both S,W are telling the true or only J is telling the true.
It shows that J in not telling the true which means he killed Cooper.
Problem 10.
A detective has interviewed four witnesses to a crime. From the stories of the witnesses the detective has concluded that if the butler is telling the truth then so is the cook; the cook and the gardener cannot both be telling the truth; the gardener and the handyman are not both lying; and if the handyman is telling the truth then the cook is lying. For each of the four witnesses, can the detective determine whether that person is telling the truth or lying? Explain your reasoning.