COT3100C01, Spring 2000
S. Lang
Solution Key to Assignment #1 (40 pts.)
1/30/2000
1.
(15 pts.) Recall the following definitions and a theorem:
Definition
. An integer
a
is even if
a
= 2
b
for some integer
b
.
(That is, there exists an
integer
b
such that
a
= 2
b
.)
Definition
. An integer
a
is odd if
a
= 2
b
+ 1 for some integer
b
.
(That is, there exists
an integer
b
such that
a
= 2
b
+ 1.)
Definition
. An integer
a
is a divisor of integer
b
, denoted
a

b
, if
a
≠
0 and there exists
integer
c
such that
b
=
ac
.
Theorem.
Each integer is either even or odd (but not both).
Theorem.
The sum of two odd integers is even.
Theorem.
If the product of two integers is even, then at least one of them is even.
Use these definitions and the theorem (and other appropriate laws) to answer each of
the following questions:
(a)
Prove that
m
is an even integer iff

m
is even.
(Part One) Prove if
m
is an even integer, then

m
is an even integer.
Since
m
is an even integer by assumption,
m
= 2
n
 (1), for some integer
n
, by
the definition of even integer.
Multiplying both sides of (1) by (

1) yields

m
=

2
n
= 2(

n
)  (2).
Since

n
is an integer, so (2) implies

m
is an even
integer, by the definition of even integer.
(Part Two) Prove if

m
is an even integer, then
m
is an even integer.
Since

m
is an even integer by assumption, then

(

m
) =
m
is an even integer from
the results of Part One.
Thus, Part Two is proved.
(b) Prove by using the definitions that the sum of an even integer and an odd integer
is odd.
Let
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 Spring '09
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