Assignment 3 Solutions
CS2102, Fall 2009
1.
(2.1, 1)
Answer: 1a. False
Explanation: Symbolically, for the menagerie set of animals
M
, the statement is
M
x
:
∃
⦁
color(x) = red. Because there are no animals in the menagerie that are
red, this statement is false.
Answer: 1b. True
Explanation: Because the menagerie consists of dogs, cats, and birds, then each
animal is either a bird or a mammal. Symbolically, this statement would be
M
x
:
∀
•
bird(x)
∨
mammal(x)
Answer: 1c. False.
Explanation: There are some animals in the menagerie that are not black, gray, or
brown. Therefore the statement that every animal is of one of these three colors is
false. Symbolically, this statement is
M
x
:
∀
•
black(x)
∨
grey(x)
∨
brown(x).
Answer: 1d. True
Explanation: There is a bird in the menagerie. A bird is not a dog or a cat.
Therefore the statement that there is an animal in the menagerie that is not a dog
or a cat is true. Symbolically, the statement is
M
x
:
∃
⦁
¬dog(x)
∧
¬cat(x).
Answer: 1e. False
Explanation: There are 5 blue birds in the menagerie. Therefore the statement that
no animal in the menagerie is blue must be false. Symbolically this statement is
M
x
:
∀
•
¬blue(x)
Answer: 1f. True
Explanation: The menagerie has 2 black dogs, ten black cats, and one black bird.
Therefore there are a dog, cat, and bird in the menagerie that all have the same
color. Symbolically, this is
M
z
y
x
:
,
,
∃
 dog(x) ^ cat(y) ^ bird(z)
⦁
(color(x) =
color(y) = color(z)).
2.
(2.1, 6)
Answer/Explanation: 6a. If m= 25 and n = 10, The predicate reads “If 25 is a
factor of 100, then 25 is a factor of 10”. 25 is a factor of 100, but it is not a factor
of 10. Therefore, the statement is false for these m and n values (recall 1
→
0 = 0).
Answer: 6b. n =100 , m = 2500
Explanation: Any set of values where m is a factor of n
2
but m is not a factor of n
is an acceptable answer here.
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View Full DocumentAnswer/Explanation: 6c. If m = 5 and n = 10, then the predicate reads “If 5 is a
factor of 100, then 5 is a factor of 10”. 5 is a factor of 100, and 5 is a factor of 10.
Since both parts of the statement are true, the statement is true (recall 1
→
1 = 1).
Answer: 6d. n = 4, m = 2, or n = 5, m = 3.
Explanation: Any n and m values where m is a factor of n
2
and n will make the
statement true. Also, the statement will be true for any value where m is not a
factor of n
2
, regardless of whether or not m is a factor of n. Recall that 1
→
1 = 1
and 0
→
1 = 1 and 0
→
0 = 1.
3.
(2.1, 10)
Answer: If a = 1, the statement is false.
Explanation: The statement says that for every integer a, (a  1)/a is not an integer.
To show that the statement is false, we only need to find one integer a where (a –
1)/a is an integer. If a = 1, (a  1)/a = (1 – 1)/1 = 0. 0 is an integer, so this is a valid
counterexample to prove that the statement is false.
4.
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 Spring '08
 KNIGHT
 Computer Science, Prime number, Rational number

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