3
IMPORTAglyph1197T
In a proof, we are given some information, and logic is employed to effect deduction,
that is, to make new conclusions from the given information.
In our example, we see that given two facts: Definition 1.1 and
8=2∙4
, it follows “logically” that
8
is
even. So there are two essential ingredients in a proof: inputted information
and logic
.
IMPORTAglyph1197T
Definitions represents a major input in a proof. There is more than one way to define
something, and our proof depends on the definitions we use
.
Example 1.3
We give an alternative definition of even integers, and a corresponding proof that
8
is even.
Definition 1.1a
An integer
ݔ
is
even
if
ݔ/2
is an integer.
Proof
Since
8/2=4
, which is an integer, we have
8
is even by Definition 1.1a.
Example 1.4
Similarly, we can disprove that
7
is even (or equivalently, to prove that
7
is not even).
Proof 1
Since
7
is between
6=2ሺ3ሻ
and
8=2ሺ4ሻ
, and there is no integer between
2
and
3
, we know
that there is no integer
݊
such that
2݊=7
. So, by Definition 1.1,
7
is not even.
Proof 2
Since
7/2=3.5
, which is not an integer,
7
is not even by Definition 1.1a.
Example 1.5
How can we define an
odd
integer?
Definition 1.2
An integer
ݔ
is
odd
if
ݔ=2݊+1
for some integer
݊
.
Definition 1.2a
An integer
ݔ
is
odd
if
ݔ
is not even.
IMPOTAglyph1197T
Definition 1.2a is legitimate because we have already defined an even integer.
Example 1.6
Prove that
7
is an odd number.
Proof 1
Since
7=2ሺ3ሻ+1
and
3
is an integer, by Definition 1.2,
7
is an odd number.
Proof 2
Since we have proved that
7
is not even, by Definition 1.2a,
7
is odd.
In Proof 2, we use a previously established fact, namely,
7
is not even (Example 1.4).
IMPOTAglyph1197T
Apart from using definitions, we often use proven theorems or results in proofs.
We move beyond “even” and “odd”, and define another fundamental mathematical concept:
divisibility
.
Definition 1.3
An integer
ܽ
divides
an integer
ܾ
(denoted
ܽ|ܾ
) if
ܾ=ܽ݊
for some integer
݊
.
Definition 1.4
An integer
ܽ
is a
divisor
(
factor
) of an integer
ܾ
if
ܽ
divides
ܾ
.
Definition 1.5
An integer
ܾ
is a
multiple
of an integer
ܽ
if
ܽ
is a divisor
of
ܾ
.
IMPOTAglyph1197T
•
“
ܽ
divides
ܾ
” is equivalent to “
ܽ
is a divisor of
ܾ
”, and “
ܾ
is a multiple of
ܽ
”.
•
We introduce a
notation
:
ܽ|ܾ
. Notations are extremely important in mathematics! We use
ܽ∤ܾ
to denote “
ܽ
does not
divide
ܾ
”, just as
≠
denotes inequality. In general, “slash” means “not”
.
Example 1.7
Suppose we have the following alternative definition of divisibility.
Definition 1.3a
An integer
ܽ
divides
an integer
ܾ
(denoted
ܽ|ܾ
) if
ܾ/ܽ
is an integer.