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Introduction
Mathematical Induction
Discrete Mathematics
Andrei Bulatov
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202
Principle of Mathematical Induction
Climbing an infinite ladder
We can reach the first rung
For all
k,
standing on the
rung
k
we can step on
the rung
k + 1
Can we reach every step of it, if
Discrete Mathematics – Mathematical Induction
203
Principle of Mathematical Induction
Principle of mathematical induction
:
To prove that a statement that assert that some property
P(n)
is
true for all positive integers
n, we complete two steps
Basis step
:
We verify that
P(1)
is true.
Inductive step
:
We show that the conditional statement
P(k)
→
P(k + 1)
is true for all positive integers
k
Symbolically,
the statement
(P(1)
∧ 2200
k (P(k)
→
P(k + 1)))
→ 2200
n P(n)
How do we do this?
P(1)
is usually an easy property
To prove the conditional statement, we assume that P(k)
is true (it
is called
inductive hypothesis
) and show that under this assumption
P(k + 1)
is also true
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204
The Domino Effect
Show that all dominos fall:
Basis Step:
The first domino falls
Inductive step:
Whenever a domino falls, its next neighbor will
also fall
Discrete Mathematics – Mathematical Induction
205
Summation
Prove that the sum of the first
n
natural numbers equals
that is
P(n):
`the sum of the first
n
natural numbers …
Basis step:
P(1)
means
Inductive step:
Make the inductive hypothesis,
P(k)
is true, i.e.
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This note was uploaded on 10/14/2011 for the course MACM 101 taught by Professor Pearce during the Spring '08 term at Simon Fraser.
 Spring '08
 PEARCE

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