4130 HOMEWORK 1
Due Thursday February 4
(1) Section 1.1.3 Exercise 2b.
“The only even prime is 2.” There are many different ways of approaching the problem. One way
is
∀
n
∈
N
(
n
is even
∧
n
is prime =
⇒
n
= 2)
.
The negation is
∃
n
∈
N
(
n
is even
∧
n
is prime
∧
n
6
= 2)
.
That is, “There exists an even prime which is not equal to 2.”
(2) Section 1.1.3 Exercise 3b.
“Every nonzero rational number has a rational reciprocal.”
∀
x
∈
Q
\ {
0
}∃
y
∈
Q
(
xy
= 1)
.
The corresponding statement with quantifiers reversed is:
∃
y
∈
Q
∀
x
∈
Q
\ {
0
}
(
xy
= 1)
.
This is false, because if
y
∈
Q
is such that
yx
= 1 for all
x
∈
Q
\ {
0
}
then
y
= 2
y
= 1 which is
impossible.
(3) Let
A
be a set and let
P
(
a
) be a statement about an element of
a
. We write
∃
!
a
∈
A P
(
a
)
for “there exists a unique
a
∈
A
such that
P
(
a
)”.
(a) Write the statement
∃
!
a
∈
A P
(
a
) in a form which uses the quantifiers
∀
and
∃
, and no connec
tives apart from
∧
,
∨
and
¬
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 '08
 PROTSAK
 Natural number, Prime number, Rational number

Click to edit the document details