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Discussion #13
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Discussion #13
Induction
(the process of deriving generalities from particulars)
Mathematical Induction
(deductive reasoning over the natural numbers)
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Topics
•
Proof by Induction
•
Summary of Proof Techniques
Discussion #13
3/18
Statements that Depend on a
Natural Number
•
Sometimes we wish to prove that a statement is true for all
natural numbers n
≥
0 or for all natural numbers greater than
some initial number.
Examples:
–
Prove: The sum of the numbers 1 to n is n(n+1)/2.
–
Prove: n
2
> 2n + 1, for n
≥
3.
–
Prove: If T is a full binary tree, then the number of nodes at each level
n of T is 2
n
.
–
Prove: If E is an expression with n occurrences of binary operators,
then E has n+1 operands.
•
Each statement S to be proved depends on a single number n.
–
We can write S(n) to denote the statement that depends on n.
–
Thus, for the first statement above, S(n) is “The sum of the numbers 1
through n is n(n+1)/2.”
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Prove S(n) for All n
≥
m
•
To prove a statement S(n) for n
≥
m, we must show
that S(m), S(m+1), S(m+2), … are all true.
–
Example: For S(n) = “The sum of the numbers 1 through n
is n(n+1)/2” we must prove:
•
The sum of the numbers 1 to 1 is 1(1+1)/2.
•
The sum of the numbers 1 to 2 is 2(2+1)/2.
•
The sum of the numbers 1 to 3 is 3(3+1)/2.
•
…
–
Unfortunately, we can never reach the end.
•
The simple pattern of one after the next, however, lets
us solve the problem.
Discussion #13
5/18
Modus Ponens is the Key
S(1)
S(1)
⇒
S(2)
S(2)
S(2)
⇒
S(3)
S(3)
…
S(k)
S(k)
⇒
S(k+1)
S(k+1)
…
If we can get started …
And then show for an arbitrary
natural number k that this
implication is a tautology …
We can conclude that S is true for
the next natural number, k+1 (if it
is true for k).
Then since k is chosen arbitrarily
(works for any natural number), S
must be true for k = 2, 3, … (all
natural numbers).
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Induction Fundamentals by Example
Prove: S(n), e.g. Prove: “The sum of the numbers from 1 to n is n(n+1)/2.”
Basis:
(i.e. this is how we get started)
S(1) holds.
i.e. The sum of the numbers from 1 to 1 is 1(1+1)/2 is a true statement.
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This note was uploaded on 03/02/2012 for the course C S 236 taught by Professor Michaelgoodrich during the Winter '12 term at BYU.
 Winter '12
 MichaelGoodrich

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