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Exercise
Set
3.1
*
1.
A menagerie consists of seven brown dogs, two black dogs,
six gray cats, ten black cats, five blue birds, six yellow birds,
and one black bird. Determine which of the following state
ments are true and which are false.
a.
There is an animal in the menagerie that is red.
b.
Every animal in the menagerie is a bird or a mammal.
c.
Every animal in the menagerie is brown or gray or black.
d.
There is an animal in the menagerie that is neither a cat
nor a dog.
e.
No animal in the menagerie is blue.
f.
There are in the menagerie a dog, a cat, and a bird that
all have the same color.
2. Indicate which
of the following statements are true and
which are false. Justify your answers as best as you can.
a.
Every integer is a real number.
b.
0 is a positive real number.
c.
For all real numbers
r,
r
is a negative real number.
d.
Every real number is an integer.
3. Let
P(x)
be the predicate
"x
>
l/x."
a.
Write
P(2),
P(~),
P(I),
P(~),
and
P(8),
and
indicate which of these statements are true and which
are false.
b.
Find the truth set of
P(x)
if the domain of
x
is R, the set
of all real numbers.
c.
If the domain is the set R+ of all positive real numbers,
what is the truth set of
P(x)?
4.
Let
Q(n)
be the predicate
"n
2
:::
3D."
a.
Write
Q(2), Q(2), Q(7),
and
Q(7),
and indicate
which of these statements are true and which are false.
b. Find the truth set of
Q(n)
if the domain of
n
is Z, the set
of all integers.
c.
If the domain is the set Z+ of all positive integers, what
is the truth set of
Q(n)?
5.
Let
Q(x,
y)
be the predicate "If
x
<
y
then
x
2
<
y2"
with
domain for both
x
and
y
being the set R of real numbers.
a.
Explain why
Q(x, y)
is false if
x
=
2 and
y
=
I.
b.
Give values different from those in part (a) for which
Q(x, y)
is false.
c.
Explain why
Q(x, y)
is true if
x
=
3 and
y
=
8.
d.
Give values different from those in part (c) for which
Q(x, y)
is true.
6.
Let
R(m, n)
be the predicate "If
m
is a factor of
n
2
then
m
is a factor of
n,"
with domain for both
m
and
n
being the set
Z of integers.
a.
Explain why
R(m, n)
is false if
m
=
25 and
n
=
10.
b.
Give values different from those in part (a) for which
R(m, n)
is false.
c.
Explain why
R(m, n)
is true if
m
=
5 and
n
=
10.
d.
Give values different from those in part (c) for which
R(m, n)
is true.
7. Find the truth set of each predicate.
a.
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This note was uploaded on 06/12/2011 for the course MATH 103 taught by Professor Wouters during the Spring '08 term at Wisc Oshkosh.
 Spring '08
 WOUTERS
 Logic, Algebra

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