Proofs
Mathematical Induction
R. Inkulu
http://www.iitg.ac.in/rinkulu/
(Mathematical Induction)
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Outline
1
Ordinary
2
Strong
3
Generalized
4
Structural
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(First) Principle of mathematical induction1
∀P [[P(1) ∧ ∀k∈N (P(k) → P(k + 1))] → ∀n∈N P(n)]
”P(1) true” is known as the basis; ”P(k) true” is termed as the induction
hypothesis; P(k) → P(k + 1) is the inductive step
a variant: ∀P [[P(j) ∧ ∀k≥j (P(k) → P(k + 1))] → ∀n≥j P(n)]
1
a.k.a. ordinary induction
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An example
If S is a finite set with n elements where n is a positive integer, then S has 2n
subsets.
(Mathematical Induction)
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Another example
Every amount of 12 cents or more can be formed using just 4-cent and 5-cent
stamps.
(Mathematical Induction)
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Correctness
well-ordering principle
⇒
first principle of mathmatical induction
(Mathematical Induction)
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Outline
1
Ordinary
2
Strong
3
Generalized
4
Structural
(Mathematical Induction)
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Second principle of mathematical induction2
∀P [[P(1) ∧ ∀k∈N (P(1) ∧ P(2) ∧ . . . ∧ P(k)) → P(k + 1))] → ∀n∈N P(n)]
a variant:
∀P [[P(j) ∧ ∀k (P(j) ∧ P(j + 1) ∧ . . . ∧ P(k)) → P(k + 1))] → ∀n≥j P(n)]
2
a.k.a. strong induction or course of values induction or complete induction
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An example
Given that every simple polygon with number of vertices greater than 3 has an
interior diagonal, a simple polygon with n sides (for n ≥ 3) can be
triangulated into n − 2 triangles.
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Another example
Every amount of 12 cents or more can be formed using just 4-cent and 5-cent
stamps.
(set yielding k + 1 cents’ stamps) = (set yielding k − 3 cents’ stamps) ∪ (1 4-cent stamp)
hence, requires four cases as part of the basis
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Correctness
well-ordering principle
⇒
second principle of mathematical induction
(Mathematical Induction)
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Outline
1
Ordinary
2
Strong
3
Generalized
4
Structural
(Mathematical Induction)
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Description
Extending ordinary/strong induction to prove results about sets that have the
well-ordering property besides N.
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An example
Suppose that am,n is defined recursively as follows.
a0,0 = 0,
am,n = am−1,n + 1 if n = 0 and m > 0,
and am,n = am,n−1 + n if n > 0.
Show that am,n = m +
(Mathematical Induction)
n(n+1)
2
for all nonegative integers m and n.
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Correctness
Let S be any set that has the well-ordering property. A lexicographic ordering
of S leads to have a bijection from N to S.
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Outline
1
Ordinary
2
Strong
3
Generalized
4
Structural
(Mathematical Induction)
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Description
Extending ordinary/strong induction to recursive structures.
(Mathematical Induction)
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An example: recursive structure of full binary tree
basis step: full binary tree comprising a single vertex
recursive step: if T1 and T2 are two disjoint full binary trees, join the
roots of T1 and T2 to a new node to get a new full binary tree, denoted
with T = T1 .T2
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An example: recursive structure of height of full binary
tree (cont)
basis step: the height of the full binary tree T consisting of only a root r
has h(T) = 0
recursive step: if T1 and T2 are full binary trees, then the full binary tree
T = T1 .T2 has height h(T) = 1 + max(h(T1 ), h(T2 ))
(Mathematical Induction)
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An example: upper bounding the number of nodes in
terms of the height of full binary tree (cont)
If T is a full binary tree, then the number of nodes in T is upper bounded by
2h(T)+1 − 1, for h(T) being the height of T.
- induction on height of the tree
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