Download Proving Universal Statements

Proving Universal Statements
The vast majority of mathematical statements to be proved are
universal. In discussing how to prove such statements, it is helpful to
imagine them in a standard form:
∀x if P (x) then Q(x)
For example,
If a and b are integers then 6a2 b is even.
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Proving Universal Statements: The Method of Exhaustion
Some theorems can be proved by examining relatively small number
of examples.
Prove that (n + 1)3 ≥ 3n if n is a positive integer with n ≤ 4.
n=1
n=2
n=3
n=4
Prove for every natural number n with n < 40 that n2 + n + 41
www
is prime.
41,43/47
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,
.
.
.
Part 1. Number Systems and Proof Techniques
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ple 4.1.6 Generalizing from the Generic Particular
Motivating
Example: “Mathematical Trick”
At some time you may have been shown a “mathematical trick” like the following. You
ask a person to pick any number, add 5, multiply by 4, subtract 6, divide by 2, and subtract
original number.
you astound
person by announcing
final
Picktwice
any the
number,
add 5,Then
multiply
by 4,thesubtract
6, divide that
by their
2, and
result was 7. How does this “trick” work? Let an empty box ! or the symbol x stand
subtract
original
The
answer
for the twice
number the
the person
picks.number.
Here is what
happens
when is
the7.person follows your
directions:
Step
Pick a number.
Add 5.
Multiply by 4.
Subtract 6.
Divide by 2.
Subtract twice the original number.
Visual Result
!
:
!|||||
!|||||
!|||||
!|||||
!|||||
!||
!||
!|||||
!|||||
!||
!|||||
||
|||||
Algebraic Result
x
x +5
(x + 5) · 4 = 4x + 20
(4x + 20) − 6 = 4x + 14
4x + 14
= 2x + 7
2
a
(2x + 7) − 2x = 7
Thus no matter what number the person starts with, the result will always be 7. Note that
the x in the analysis above is particular
(because it represents a single quantity), but it
Part 1. Number Systems and Proof Techniques
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Generalising from the Generic Particular
The most powerful technique for proving a universal statement is
one that works regardless of the choice of values for x.
To show that every x satisfies a certain property, suppose x is a
particular but arbitrarily chosen and show that x satisfies the
property.
?iiT,ffrrrX+b+XHBpX+XmFfFQM2pf*PJSRyN
Part 1. Number Systems and Proof Techniques
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af
Method of Direct Proof
Express the statement to be proved in the form
“∀x, if P (x) then Q(x).”
(This step is often done mentally.)
Start the proof by supposing x is a particular but arbitrarily
chosen element for which the hypothesis P (x) is true.
(This step is often abbreviated “Suppose P(x).”)
Show that the conclusion Q(x) is true by using definitions,
previously established results, and the rules for logical inference.
?iiT,ffrrrX+b+XHBpX+XmFfFQM2pf*PJSRyN
Part 1. Number Systems and Proof Techniques
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Prove that the sum of any two even integers is even
Fm
if
n
,
mound
are
n
integers
even
the
-
Mtn
¥
is
Suppose
even
.
÷
misanwennkgf
Then
M
h
-
=
2k
Ll
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is
h
and
for
to
some
some
integer
an
integer
even
integer
L
Part 1. Number Systems and Proof Techniques
.
K
.
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Mt
n=LKt2l=2(
Therefore
,
man
is
ktl )
W
even
.
is
the
that
Prove
of
sun
odd
two
any
integers
even
.
fm
,n
if
Mand
then
n
is
Mtn
odd
are
integers
even
1¥
Suppose that
m=2K
Then
n=
Manda
't
,
Mint
$kH)t
for
,
Qltl
( zltl )
=2(
odd
are
for
-
ntegas
some
IKH +4+1
Kt
K
integer
some
integer
=
.
l
2144+2
=
Prove that every integer is rational
Y
if
n
h
-
is
itegar
an
rational
then
h
B
-
Proof
=
that
ppm
Then
So
n
h
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-
n
is
an
Meger
hp
is
rational
.
Part 1. Number Systems and Proof Techniques
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Prove that the sum of any two rational numbers is rational
F
r
,q
if
then
✓
and
rtq
q
rational
is
Prof
Suppose
Their
that
r=F
✓
,
some
q=Ee for
,
r+q=m*±e=
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Eek
.
garnration
and
for
rational
are
integers
some
.
na
min
nkgesk
Is
,
e
lto
sina.mn?Eyo?noie
Part 1. Number Systems and Proof Techniques
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Prove that the product of any two rational numbers is
rational
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Part 1. Number Systems and Proof Techniques
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Prove that the double of a rational number is rational
¥
fs
s
then
rational
in
Ls
rash ,
is
1¥
Suppose
Since
the
2
that
rational
is
previous
,
statement
=
rational
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is
s
fational
by
2.5
is
.
Part 1. Number Systems and Proof Techniques
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Prove for all integers n, if n is even then n2 is even
?iiT,ffrrrX+b+XHBpX+XmFfFQM2pf*PJSRyN
Part 1. Number Systems and Proof Techniques
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Prove By Cases: Combine Generic Particulars and Proof By
Exhaustion
Statement: For all integers n, n2 + n is even
Case 1: n is even
th
h4n=
then
4k
'
n=LlH
-
4l'
4l
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nz
that
'
+
(
et 2
-
942 abets
Then
.
4k
Lk
t
( atD=
Case 2: n is odd
htth
ht
Them
2k
+
-
HED
2/4 't
Let
Part 1. Number Systems and Proof Techniques
'
'
LKU'#
-
Hiya
1
=
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