CS0441 Discrete Structures
Recitation 2
Xiang Xiao
Section 1.1 Q14
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
C. To get an A in this class, it is necessary for you to get an A on the final.
r p
“q is necessary for p” equals to “p -> q”
Q22 a. It is necessary to wash the boss’s car to get promoted.
If you get promoted, then you must have washed the boss’s car
Section 1.1 Q22
Write each of these statements in the form “if p, then q” in
English.
c. A sufficient condition for the warranty to be good is that you
bought the computer less than a year ago.
p is sufficient for q: p -> q
If you bought the computer less than a year ago, then the
warranty is good.
e. You can access the website only if you pay a subscription fee
p only if q: p -> q
If you can access the website, then you must have paid a
subscription fee.
Section 1.1 Q22
Write each of these statements in the form “if p, then q” in
English.
g. Carol get seasick whenever she is on a boat.
q whenever p: p -> q
If Carol is on a boat, then she will get seasick.
Q28 b. I go to the beach whenever it is a sunny summer day.
Converse: If I go to the beach, then it is a sunny summer day.
Contrapositive: If I don’t go to the beach, then it is not a sunny summer day.
Inverse: If it is not a sunny summer day, then I won’t go to the beach.
Section 1.3 Question 12
Show that each of these conditional statement is a tautology
without using truth tables
a. [p  ( p  q)]  q
[p  ( p  q)]  q  [p  ( p  q)]  q
 [(p )  ( p  q)]  q
(De Morgan's)
 [ p  ( p  q)]  q
(Double Negation)
 [ p  (p  q)]  q
(De Morgan's)
 [( p  p )  ( p  q)]  q
(Distributive)
 [T  ( p  q)]  q
(Negation)
 ( p  q )  q
(Identity)
 p  ( q  q )
(Associative)
 p T
(Negation)
T
(Domination)
Section 1.3 Question 12
Show that each of these conditional statement is a tautology
without using truth tables
b.
[( p  q)  ( q  r )]  ( p  r )
Section 1.3 Q22
Show that ( p  q)  ( p  r ) and p  ( q  r ) are logically
equivalent.
( p  q )  ( p  r )  ( p  q )  ( p  r )
 ( p  q )  ( p  r )
 p  ( q  r )
 p  (q  r)
Section 1.4 Q 10
Let C(x) be the statement “x has a cat”, let D(x) be the statement
“x has a dog”, and let F(x) be the statement “x has a ferret”.
Express statements in terms of C(x), D(x) and F(x).
d. No student in your class has a cat, a dog, and a ferret.
x (C ( x )  D ( x )  F ( x )) 
x ((C ( x )  D ( x )  F ( x ))) 
x (C ( x )  D ( x )  F ( x ))
Section 1.4 10
Let C(x) be the statement “x has a cat”, let D(x) be the statement
“x has a dog”, and let F(x) be the statement “x has a ferret”.
Express statements in terms of C(x), D(x) and F(x).
e. For each of the three animals, cats, dogs, and ferrets, there is
a student in your class who has this animal as pets
x (C ( x ))  x ( D ( x ))  x ( F ( x )) 
xyz (C ( x )  D ( y )  F ( z ))
Section 1.4 Q28
Translate each of the statement into logical
expressions.
b. All tools in the correct place and are in excellent conditions.
Predicates (Domain: all things):
T(x): x is a tool; C(x): x is in the correct place;
E(x): x is in excellent conditions
x (T ( x )  (C ( x )  E ( x ))
x (T ( x )  C ( x )  E ( x ))
Correct
Incorrect
Section 1.4 Q28
Translate each of the statement into logical
expressions.
d. Nothing is in the correct place and is in excellent condition
Predicates (Domain: all things):
T(x): x is a tool; C(x): x is in the correct place;
E(x): x is in excellent conditions
x ((C ( x )  E ( x ))) 
x (C ( x )  E ( x ) 
x (C ( x )  E ( x ))