In translating each of the following to symbolic form, the word
``or'' would be replaced by Ú or \veebar . What is the
appropriate choice in each case? Explain why your choice is correct.
 To take discrete mathematics, you must have taken calculus or
a course in computer science.
Answer: Ú since it is
fine if you have taken both.
 Dinner for two includes two items from column A or three items
from column B.
Answer: \veebar since if you choose two
from A you cannot in addition choose three from B.
 When you buy a new car from Acme Motor Company, you get $
2000 back in cash or a 2% car loan.
Answer: \veebar
since you get one or the other but not both.
 School is closed if more than 2 feet of snow falls or if the
wind chill is below 50.
Answer: Ú since school is
closed if both happen to be the case.
Determine whether
is a tautology.
Answer:
p  q  ( ~ q  Ù  (p ® q))  ®  ~ p 
T  T  F  F  T  T  F 
T  F  T  F  F  T  F 
F  T  F  F  T  T  T 
F  F  T  T  T  T  T

Yes it is a tautology.
Does p ® q logically imply (p Ùq) Ú ~ p ? Justify your answer.
Answer:
p  q  p ® q  (pÙq)  Ú  ~ p 
T  T  T  T  T  F 
T  F  F  F  F  F 
F  T  T  F  T  T 
F  F  T  F  T  T 
Yes. Whenever p® q is T, so is (pÙq) Ú ~ p.
Use truth tables to show that both
are equivalent to p.
Answer:
p  q  p  Ú  (pÙq)  p  Ù  (p Úq) 
T  T  T  T  T  T  T  T 
T  F  T  T  F  T  T  T 
F  T  F  F  F  F  F  T 
F  F  F  F  F  F  F  F

Determine whether
(p Úq) ® r and p Ú(q ® r ) 

are equivalent.
Answer:
p  q  r  (pÚq)  ®  r  p  Ú
 (q® r) 
T  T  T  T  T  T  T  T  T 
T  T  F  T  F  F  T  T  F 
T  F  T  T  T  T  T  T  T 
T  F  F  T  F  F  T  T  T 
F  T  T  T  T  T  F  T  T 
F  T  F  T  F  F  F  F  F 
F  F  T  F  T  T  F  T  T 
F  F  F  F  T  F  F  T  T

Their truth tables are different. They are not equivalent.
 Determine whether
(p « q) « r and p « (q « r ) 

are equivalent.
 Is ((p « q) « r) « (p « (q « r )) a tautology?
Explain.
Answer:
(a) They are equivalent. Look at their truth tables:
p  q  r  (p « q)  «  r  p  «  (q « r) 
T  T  T  T  T  T  T  T  T 
T  T  F  T  F  F  T  F  F 
T  F  T  F  F  T  T  F  F 
T  F  F  F  T  F  T  T  T 
F  T  T  F  F  T  F  F  T 
F  T  F  F  T  F  F  T  F 
F  F  T  T  T  T  F  T  F 
F  F  F  T  F  F  F  F  T

(b) Look at the truth table above. ((p « q) « r) « (p « (q « r )) is a tautology.
It
always has truth value T.