Therefore the Fibonacci numbers in this sum are all different from
F
l
. As a result, one can write
m
as
the sum of one
F
l
, and a few distinct Fibonacci numbers which are less than
F
l
. We finished the induction.
Lecture 01
§
3 Definition
•
Definition (in mathematics) gives a precise meaning of a term.
•
Some typical definition forms:
An object
X
is called
the term being defined
provided it satisfies
specific conditions
.
We say
X
is
the term being defined
provided it satisfies
specific conditions
.
•
Examples:
(1) Define the following terms: divisible, odd, even, prime, composite. You could use the
basic concept of integers (including additions and multiplications).
(2) Suppose the distance between two points in the plane is defined already. Give a defi
nition of
collinear
of three points.
§
4 Theorem
•
A
theorem
is a declarative statement (about mathematics) for which there is a proof.
•
Differences between theorems, conjectures, and mistakes.
•
The table of “If
A
then
B
” statement
A
B
True
True
Possible
True
False
Impossible
False
True
Possible
False
False
Possible
•
Equivalent expressions to “If
A
then
B
”:
(1)
A
implies
B
.
(2) Whenever
A
, we have
B
.
(3)
A
is sufficient for
B
.
(4)
B
is necessary for
A
.
(5)
A
=
⇒
B
.
•
(Vacuous truth) Sometimes the condition
A
is always impossible. In this case the statement
“If
A
then
B
” is always true.
•
The table of “
A
if and only if
B
” statement
A
B
True
True
Possible
True
False
Impossible
False
True
Impossible
False
False
Possible
•
Equivalent expressions to “
A
if and only if
B
”:
(1)
A
iff
B
.
(2)
A
is sufficient and necessary for
B
.
1
(3)
A
is true exactly when
B
is true.
(4)
A
⇐⇒
B
.
•
And, Or, and Not statements.
§
5 Proof
•
Direct proof of “if
A
then
B
”
(1) Invent suitable notations and restate
A
.
(2) Write down
B
.
(3) Figure out a bridge between
A
and
B
.
•
Direct proof of “
A
if and only if
B
”: prove “if
A
then
B
” and “if
B
then
A
”.
•
Examples:
(3) Let
x
be an integer. If
x >
1, then
x
3
+ 1 is composite.
(4) Suppose
x, y, z
are positive real numbers. Prove that
x
3
+
y
3
+
z
3
≥
3
xyz
.
HW1(a)
(Due 2/1/2016)
•
3.4
•
4.2(a),(g),(h)
•
5.21
2
Lecture 02
§
6 Counterexample
•
To disprove “if
A
then
B
”, we just need to find an example where
A
is true but
B
is false.
•
Examples:
(1) Disprove: If
a
,
b
,
c
are positive integers such that
a

bc
, then
a

b
or
a

c
.
(2) Disprove: if
p
is prime, then 2
p

1 is also prime.
§
7 Boolean algebra
•
Boolean algebra includes expressions containing letters and operations, where each letter
stands for the value TRUE or FALSE. The basic operations are
∧
and
:
x
∧
y
is true if and only if both
x
and
y
are true.
∨
or
:
x
∨
y
is true if and only if at least one of
x
and
y
is true.
¬
not
:
¬
x
is true if and only if
x
is false.
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 Fall '16
 Sam Liu
 Natural number, Equivalence relation