Constructive vs Non-Constructive Existence Proof
One way to prove
∃x Q(x)
is to find an x in that makes Q(x) true.
Sometimes we do not find an x that makes Q(x) true
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Part 1. Number Systems and Proof Techniques
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Show that ∃ irrational numbers x and y such that xy is
rational
L
Case
Let
Xtr
is
y=E
rationainumbwng
Xt=@@
→
irrational
Case 2
:
if
Fit
tr
?iiT,ffrrrX+b+XHBpX+XmFfFQM2pf*PJSRyN
is
irrational
XEKRFERY
X=@
Part 1. Number Systems and Proof Techniques
number
then
wf
'Iri=2
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Mathematical Induction
Mathematical induction is one of the more recently developed
techniques of proof in the history of mathematics.
It is used to check conjectures about the outcomes of processes
that occur repeatedly and according to definite patterns.
In general, mathematical induction is a method for proving that a
property defined for integers n is true for all values of n that are
greater than or equal to some initial integer
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Part 1. Number Systems and Proof Techniques
)
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Example: Domino Effect
One domino for each natural number, arranged in order.
I will push domino 0 (the one at the front of the picture)
towards the others.
00
For every natural number m, if the m’th domino falls, then the
(m + 1)st domino will fall.
Conclude: All of the Dominoes will fall.
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Part 1. Number Systems and Proof Techniques
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#
DDB
Proving by induction that a property holds for every natural
number n
Prove that the property holds for the natural number n = 0.
→
.es
Prove that if the property holds for n = m (for any natural
number m) then it holds for n = m + 1.
The validity of proof by mathematical induction is generally taken as
an axiom. That is why it is referred to as the principle of
mathematical induction rather than as a theorem.
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Part 1. Number Systems and Proof Techniques
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Pre
f
m
if
the
=
the
property
holes
property
for
holds
for
M
then
=
my
A proof of a property by induction looks like this
Base Case:
Show that the property holds for n = 0.
Inductive Step: Assume that the property holds for n = m.
Show that it holds for n = m + 1.
Conclusion: You can now conclude that the property holds for
every natural number n.
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Part 1. Number Systems and Proof Techniques
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ITLt3
.
.
.
+50+5
't
.
.
.
:⇒FE¥¥T
¥43.7
'
-110g
.÷
!oIII÷n=n¥101
101
.
50
Prove that
lt2+
.th=hlh¥
.
#
't
-
Bak
case
:
h¥
#
Induction
It
V.
sleep
n
1+2+3
't
.
.+n=
1¥
MID
✓
1
then
1t2t3t.tnt(hH)=(htDK=h£
Assume
1+2+3
+
.
that
.int
Itt
1)
@
+
.
=
.tn
nh¥
=
NIH
+
1)
µ
+
NI'44¥=
=
't
=
M¥+24 .ch#h+2)=
@
D ((
=
ntiki )
.
Example: Proof by Induction
For every natural number n,
0 + 1 + ··· + n =
n(n + 1)
.
2
Base Case: Take n = 0. The left-hand-side and the
right-hand-side are both 0 so they are equal.
Inductive Step:
Assume that the property holds for n = m, so
0 + 1 + ··· + m =
m(m + 1)
.
2
Now consider n = m + 1. We must show that
0 + 1 + · · · + m + (m + 1) =
?iiT,ffrrrX+b+XHBpX+XmFfFQM2pf*PJSRyN
(m + 1)(m + 2)
.
2
Part 1. Number Systems and Proof Techniques
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Proof continued
Since
0 + 1 + ··· + m =
m(m + 1)
.
2
m(m + 1)
+m+1
2
m(m + 1) + 2(m + 1)
=
2
(m + 1)(m + 2)
=
2
0 + 1 + · · · + m + (m + 1) =
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Part 1. Number Systems and Proof Techniques
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1+3+5+7
K=o
/t3t5t
.
?
@kt )
1
1<=1
1+3=4
1<=2
K=
+9=7
1+3+5=9
1+3+5+7=16
]
K=y
ltst5t7
Maybe
9=25
+
1+3+5-1
.
...
+
( 2kt )=
(Kty
'
?
Proof
induction
mathematical
by
.
=
Baa
case
Induction
if
1<=0
:
the
12=1
sleep :
FK
1+7
By
( tot
1¥
principle of
Its
+
'
.
.
.
.
.
.+fkH
kkttt
mathematical
=
( KHI
( It KAHD
idnctron
'
then
( ( KHHT
=
It
+
.
.
Key
=
#p