MathematicInduction
YanHuang
Objective
• InductiononIntegers
• InductiononStructures
Prove
% %+1
1+2+⋯+% =
,
2
∀% ∈ ℕ
ℕ:thesetofnaturalintegers,{1,2,3,…}
∀% ∈ ℕ:foreveryinteger% inℕ.
./0
, - = 1 + 2 + ⋯+ %
./1
Firstattempt
Prove
% %+1
1+ 2+⋯+% =
,
2
∀% ∈ ℕ
%=1
%=2
%=3
%=4
⋯⋯
Youwillneverfinishtheproof…!
MathematicInduction
• Basestep
- Provetheidentityforaparticular% value(suchas% = 1,
dependingonyourgoal,).
• Inductivehypothesis
- Assumingtheidentityholdsforall% ≤ 5,
• Inductionstep
- Provetheidentityalsoholdsfor% = 5 + 1.
Prove
% %+1
1+ 2+⋯+% =
,
2
Basestep:
Inductivehypothesis:
Inductionstep:
∀% ∈ ℕ
Arewedone?
• Yes!Butwhy?(Ihaven’tproved
thetheoremformany
particular% suchas
383210348 yet.AmIreally
done?)
Infinity?
• Doesthisproofshowthat1 + 2 + ⋯ + % =
when% = ∞?
0 081
9
even
Prove∀% ∈ ℕ,
0
9
9
9
1 +2 +⋯+% =
081 9081
;
.
∀% ∈ ℕ means“foreveryinteger% in{1, 2, 3, … }.”
Basestep:
Prove∀% ∈ ℕ,
0
9
9
9
1 +2 +⋯+% =
081 9081
;
.
Inductivehypothesis:
Thereexistssome5 suchthatif% ≤ 5,
9
9
9
1 +2 +⋯+% =
Inductionstep:
0 081 9081
;
.
Prove∀% ∈ ℕ,
0
9
9
9
1 +2 +⋯+% =
Inductionstep:
If% = 5 + 1,
081 9081
;
.
Prove∀% ∈ ℕ,
Basestep:
%! ≤ %0 .
Prove∀% ∈ ℕ,
Inductivehypothesis:
Inductionstep:
%! ≤ %0 .
InductionoverStructures
StructuralInduction
1. Provethestatementforthebasecases.
2. Statethehypothesis
3. Provethestatementforeveryinductiverules
StructuralInduction
Theset@ isdefinedasfollows,
(1)3 ∈ @.
(2)If A, B ∈ @,then A + B ∈ @.
Prove@ ⊆ 3% % ∈ ℤ8 }.
BaseStep
Theset@ isdefined asfollows,
(1)3 ∈ @.
(2)If A, B ∈ @,then A + B ∈ @.
Prove@ ⊆ 3% % ∈ ℤ8 }.
• Wewanttoshow3 ∈ {3%|% ∈ ℤ8 }.
Proof:
Let% = 1 ∈ ℤ8,3 = 3 ∗ 1 = 3% ∈ {3%|% ∈ ℤ8 }.
Thus,3 ∈ {3%|% ∈ ℤ8 }.
InductiveHypothesis
Theset@ isdefined asfollows,
(1)3 ∈ @.
(2)If A, B ∈ @,then A + B ∈ @.
Prove@ ⊆ 3% % ∈ ℤ8 }.
If A, B ∈ @,thenA, B ∈ {3%|% ∈ ℤ8 }.
InductionStep
Theset@ isdefined asfollows,
(1)3 ∈ @.
(2)If A, B ∈ @,then A + B ∈ @.
Prove@ ⊆ 3% % ∈ ℤ8 }.
• Wewanttoshow:
If A, B ∈ @,then A + B ∈ {3%|% ∈ ℤ8 }.
Proof:
SinceA, B ∈ @,bytheinductivehypothesis,A, B ∈ {3%|% ∈
ℤ8 }.Hence,thereexist%1 , %9 ∈ ℤ8 suchthatA = 3%1 , B =
3%9 .Therefore,A + B = 3 %1 + %9 ∈ {3%|% ∈ ℤ8 } because
%1 + %9 ∈ ℤ8.
Exercise
• Prove∀% ∈ ℕ,
081
3
– 3
9
G
H
0
3 + 3 + 3 + 3 +…+ 3 =
2