CS 173: Discrete Structures, Spring 2010
Homework 2 Solutions
This homework contains 6 problems worth a total of 50 points.
1.
[9 points] Logic notation
Translate the following sentences into propositional or predicate logic. Use the short
hand symbols (e.g.
_
) and de ne the meaning of each of your predicates and proposi
tional variables. (See pages 11 and 4243 of Rosen for examples.) Be sure to include
a domain (aka replacement set) for each quanti ed variable. You may need to rewrite
the sentences slightly so as to make a variable more explicit.
(a) Every cookie that contains lots of butter is tasty but is also fattening.
[Solution]
8
x
2
C; B
(
X
)
!
(
T
(
x
)
^
F
(
x
))
where
C
is set of cookies, and
B
(
x
) means
x
contains lots of butter,
T
(
x
) means
x
is tasty, and
F
(
x
) means
x
is fattening.
(b) Either the tablecloth is pink, or the tablecloth is white and the red light is shining
on it.
[Solution]
p
_
(
w
^
r
)
where
p; w; r
represent ‘The tablecloth is pink’, ‘The tablecloth is white’, and ‘The
red light is shining on the tablecloth’ respectively.
Note:
A solution using
instead of
_
is also correct.
(c) Exactly one integer is equal to its own negation.
[Solution]
9
x
2
Z
;
8
y
2
Z
;
(
x
=
x
)
^
[(
x
6
=
y
)
!
(
y
6
=
y
)]
where
Z
denotes the set of integers. This solution states that there exists at least
one integer,
x
, equal to its own negation and for all integers
y
6
=
x
,
y
is not equal
to its own negation, therefore
x
is unique. A solution using
9
! is also acceptable,
e.g.
9
!
x
2
Z
(
x
=
x
)
For example, the sentence \Every car that is driven in Berkeley is a hybrid or runs on
biodiesel." might be rendered as
8
x
2
C; D
(
x
)
!
(
H
(
x
)
_
B
(
x
))
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 Spring '08
 Erickson

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