to
:
PROOF
TECHNIQUES
TOPICS ;
•
COUNTEREXAMPLES
•
DIRECT
•
PROOF
•
PROOFS
BY
EXAMPLES :
CONTRAPOSITIVE
DIVISIBILITY
* PRE CALL VIDEO
:
ODD) EVEN
&
EUCLID'S
LEMMA
/ PRIMES / ETC
.
,
DIVISIBILITY
counterexamples
ONE
*
WAY
TO
PROVE THAT A
"
"
FOR ALL
STATEMENT
IS
FALSE
'
y
ALWAYS
*
EI
.
ALL
EXPLAIN
PROFESSORS
EVERY
For
WHY
PRIME
ALL
YOUR
HAVE
COUNTEREXAMPLE
BLONDE
NUMBER IS
REAL NUMBERS
ODD
X
HAIR
IS
A
COUNTEREXAMPLE
.
.
AND
y
,
Vxty
=
Tx
try
.
.
SOMETIMES
EI
.
YOU
MIGHT
HAVE TO
DO
SOME
,
For ALL
POSITIVE
INTEGERS
n
,
EXPERIMENTATION
THE NUMBER
TO
't ht 17
n
FIND
IS
ONE :
PRIME
.
DIRECTPROOFSTX
ONE
WAY
VERIFY
MANY
TO
IT
PROVE
A STATEMENT
METHOD
STATEMENTS
THAT
DIRECT
OF
SUCH A
2. CONCLUDE
P
THAT
'
"
P
STATEMENT
1. ASSUME THAT
PROOF
IS
TO
,
THEN
DIRECTLY
IS TRUE
Q
IS
Q
TRUE
.
:
.
JUST DIRECTLY
!
YOU 'LL NEED TO PROVE ARE
IF
PROVE
true
"
.
"
TO
IS
IMPLICATIONS :
EI
.
IF
m
AND
n
ARE
ODD
INTEGERS
,
THEN
mtn
IS
EVEN
.
DEAUNGWITHQVANTIFIERSINDIRELTPROITWOTYPES.it
(
EXISTENTIAL
.
"
*
To
PROVE
"
"
THERE
EXISTS
THERE EXISTS
/
ETC
.
)
"
SUCH THAT
-
UP
WITH A
SPECIFIC
THE STATEMENT IS TRUE
.
X
THERE
"
IS
,
COME
EI
/
"
THERE
THERE
EXISTS
IS
AN
A
X
THEN SHOW THAT
,
.
BIRD
INTEGER
CANNOT
FLY
SUCH THAT
NZ
THAT
n
.
=
4
.
THE
REST
OF
DEAUNbWITHQVANTlFlERSlNDlRELTPR0#
TYPES :
TWO
2. UNIVERSAL
* TO
-
-
PROVE
(
"
"
"
FOR ALL
/
"
FOR ANY
"
/
"
EVERY
ALL
.
)
X
,
-
POSSIBILITIES FOR
FINITELY MANY
IF
INFINITELY MANY POSSIBILITIES FOR
SHOW
/ ETC
"
FOR
IF
THEN
"
THAT
THE REST
X
:
OF THE
PROOF BY
X:
EXHAUSTION ICASES
CHOOSE A
STATEMENT
GENERIC
IS
TRUE
×
,
PROOFBYEXHAVSTIONICASESEI
.
EVERY
DAY
FOR ALL
OF
THE WEEK
POSITIVE
ENDS
INTEGERS
n
IN
? 4
-
DAY
IN
ENGLISH
.
(http 33
"
,
WE
HAVE
.
INFINITELYMANYPOSSIBILITIESEI
.
For
ALL
REAL
NUMBERS
X
AND
y,
x' ty'
3
2xy
.
* THESE
APPROACHES
CAN BE
COMBINED
'
.
FOR
EXAMPLE
,
TO
PROVE
"
FOR ALL
X
s
I
CHOOSE
.
z
3
.
.
A
THERE EXISTS
GENERIC
X
( IF
YOU'VE ALREADY
SHOW
THAT P
IS TRUE
P
11
.
FINITELY MANY POSSIBILITIES :
COME UP WITH A SPECIFIC
THAT
y
SUCH THAT
y ( POSSIBLY
CHOSEN
)
DEPENDING
CASES
ON
THE
)
X
EI
.
FOR ALL NONZERO
REAL NUMBERS
NUMBER
THAT
y
SUCH
Xy
=
2
X
,
.
THERE EXISTS
A REAL
ANOTHER COMMON EXAMPLE
*
TO
:
PROVE
"
'
FOR ALL
THERE EXISTS
X
,
I
2
CHOOSE
.
.
A
YOU'VE ALREADY
3. ASSUME
P
SHOW THAT
y
IF
P
THEN
FINITELY MANY POSSIBILITIES :
y ( POSSIBLY
CHOSEN
Q
,
DEPENDING
THE
ON
)
IS TRUE
IT
FOLLOWS THAT
Q
IS
TRUE
.
.
"
"
E N
-
AND
E J
-
"
PROOFS
:
MORE
ON
THESE
LATER
!
1
.
CASES
.
"
EI
( IF
X
COME UP WITH A SPECIFIC
THAT
y
GENERIC
SUCH THAT
)
X
THECONTRAPOSITIVERECALL
:
P
IF
Q
AND
"
"
OF
"
*
STATEMENTS
ARE
IF
THEN
P
,
,
THEN THE
EONTRAPOSITIVE
"
Q
IF
is
NOT
Q,
THEN NOT
P
.
"
PROOF
OF
BY
THE
CONTRAPOSITIVE
IF
THEN
P
,
CONTRAPOSITIVE
YOU
CAN
PROVE
A
STATEMENT
"
"
FORM
:
Instead !
Q
BY
"
PROVING
ITS
"
EI
.
IF
n2
IS
EVEN
,
THEN
n
IS
EVEN
.
EI
.
FOR
ANY
INTEGERS
m
AND
n
,
IF
mtn
311
,
THEN
m
? 6
OR
n
36
.
MOREEXAM-PL.ES :
RECALL
AN
:
IF
INTEGER
*
IN
(r
EI
THIS
IF
.
a
=
"
bg
,
CASE
fb
,
IS
,
IS
SAID TO
,
WE
"
5/1
)
12
:
THAT
b= g. at
THEN
BUT
SAY
DIVIDE
INTEGER
INTEGER
AN
REMAINDER
3/12
"
A
IS ANOTHER
THERE
( 1. E.
*
DIVISIBILITY
r
q
,
'
q
b
s
IS
FOR
,
'
"
AN
INTEGER b
QUOTIENT
"
b
SUCH THAT
"
g- a
=
)
"
DIVISIBLE
SOME
BY
INTEGERS
q
"
a
.
AND
a
r
/
"
b
EI
.
PROVE
THAT
IF
41
a
,
THEN
4
/(
3 at
8
)
.
EI
.
PROVE
"
(a)
IF
"
(b)
IF
OR
DISPROVE
a/b
AND
a/b
AND
:
bla
b/c
THEN
,
THEN
,
a
-
-
b
ale
"
.
"
.
EI
.
IF
3
In
'
THEN
,
31
n
.
PREVIOUS
MORE
A
TIM (
EI
.
:
GENERAL
EUCLID'S
ARE
b
EXAMPLE
2/64
3
RESULT
LEMMA
INTEGERS
3/12
IF
)
.
AND
In
'
,
THEN
IS
TRUE :
FOR
ANY
p
lab
,
3
In
.
PRIME
NUMBER
THEN
EITHER
P,
p
la
IF
or
a
Awp
pl
b
Qi
DIE
HOW
THE
.
WRITTEN
BOIH
HIM
a
.
"
GREATEST
gcd(
AND
a
,
b)
THIS ?
COMMON
DIVISOR
"
OF
TWO
INTEGERS
a
AND
b
,
,
THE
IS
LARGEST
INTEGER
DIVIDES
THAT
b.
( B. E' 2-OUT'S
EXIST
EI
PROVE
YOU
DO
INTEGERS
IDENTITY )
m
god ( 12,20 )
=
AND
INTEGERS
FOR ANY
.
n
SUCH
AND
THAT
-
-
a
god ( b)
a
,
12 t
-
-
b
AND
20
=
=
,
THERE
mat
n
b
.
god ( 12,20 )
TIM (
b
EUCLID'S
ARE
LEMMA
INTEGERS
)
.
AND
FOR
p
ANY
lab
,
PRIME
NUMBER
THEN
EITHER
P,
p
la
IF
or
a
AND
pl
b