COT3100C01, Fall 2000
Assigned: 9/05/2000
S. Lang
Assignment #1 (40 pts.)
Due: 9/14 in class by 1:10 pm
Instructions:
Write your answer neatly and concisely.
All proofs need to be justified by using the
appropriate definitions, theorems, and logical reasoning.
Illegible scribbles or unclear logic will
result in minimum credit.
1.
(16 pts.) Recall the following definitions and theorems about integers:
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.
Theorem.
If
a

b
and
b

c
, then
a

c
.
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 Fall '09
 Logic, integer b., Definition. An integer

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