CS70 Discrete Mathematics and Probability, Fall 2009
Homework 1 Solutions
Note: These solutions are not necessarily model answers. Rather, they are designed to be tutorial in nature,
and sometimes contain a little more explanation than an ideal solution. Also, bear in mind that there may
be more than one correct solution. The maximum total number of points available is 40.
1. [Classical Logic]
[15 pts]
(a) It is useful to rephrase the sentences to be in the form “If
X
, then
Y
.” Writing sentences as quantiﬁed
propositions:
(I) “No shark ever doubts that he is well ﬁtted out.”
This means: “If a ﬁsh is a shark, then it does not doubt that it is well ﬁtted out.”
∀
x,B
(
x
)
⇒ ¬
P
(
x
)
.
[1
pt
]
(II) “A ﬁsh, that cannot dance a minuet, is contemptible.”
∀
x,
¬
U
(
x
)
⇒
F
(
x
)
.
[1
pt
]
(III) “No ﬁsh is quite certain that it is well ﬁtted out, unless it has three rows of teeth.”
This statement means that “A ﬁsh is certain that it is well ﬁtted out only if it has three rows of teeth,”
or equivalently, “If a ﬁsh is certain that it is well ﬁtted out, then it has three rows of teeth.” Thus we
have
∀
x,
¬
P
(
x
)
⇒
O
(
x
)
.
[1
pt
]
(IV) “All ﬁshes, except sharks, are kind to children.”
This means: “If a ﬁsh is not a shark, then it is kind to children.”
∀
x,
¬
B
(
x
)
⇒
N
(
x
)
.
[1
pt
]
(V) “No heavy ﬁsh can dance a minuet.”
This means: “If a ﬁsh is heavy, then it cannot dance a minuet.”
∀
x,K
(
x
)
⇒ ¬
U
(
x
)
.
[1
pt
]
(VI) “A ﬁsh with three rows of teeth is not to be despised.”
∀
x,O
(
x
)
⇒ ¬
F
(
x
)
.
[1
pt
]
Many people got the wrong order of implication in parts I, III, and V, or got the negations wrong.
Note that the contrapositive of each implication above is also an acceptable answer.