•
Basic definitions:
The terms “even integer”, “odd integer”, “perfect square”, “divisible”, “rational”, and “irrational”
are defined in the table below.
These are the definitions you should work with in the problems below.
Below,
n
always denotes an integer,
d
denotes a non-zero integer, and
x
denotes a real number.
n
even
⇐⇒
there exists
k
∈
Z
such that
n
= 2
k
n
odd
⇐⇒
there exists
k
∈
Z
such that
n
= 2
k
+ 1
n
perfect square
⇐⇒
there exists
k
∈
Z
such that
n
=
k
2
n
divisible by
d
⇐⇒
there exists
k
∈
Z
such that
n
=
dk
x
rational
⇐⇒
there exist
p, q
∈
Z
with
q
6
= 0 such that
x
=
p/q
x
irrational
⇐⇒
x
is not rational
Note that “even” is equivalent to “divisible by 2”.
(This follows immediately from the definitions of “even” and
“divisible by
d
”.)
•
Fact about even and odd integers:
We
assume
here the following result:
An integer is either even, or odd, but not
both.
In other words, this says that
“odd” is the negation of “even”
. This may seem obvious, but it is in fact a
non-trivial result that we will prove later (quite easily) using number-theoretic techniques and results (specifically, the
division algorithm).
For now, we simply assume this fact.
1

Math 347
Worksheet on “Even/odd” Proofs
A.J. Hildebrand
Even/odd proofs: Practice problems
The problems below illustrate the various proof techniques: direct proof, proof by contraposition, proof by cases, and proof
by contradiction (see the separate handout on proof techniques).
For each of these proof techniques there is at least one
problem for which the technique is appropriate. For some problems, more than one approach works; try to find the simplest

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- Addition, Rational number, Parity