Mathematical Induction and Recursion
CS 30 : Discrete Mathematics for Computer Science First Semester, AY 2011-2012
https://sites.google.com/a/dcs.upd.edu.ph/nhsh_classes/cs 30
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Nestine Hope S. Hernandez Algorithms and Complexity Laboratory Department of Computer Science University of the Philippines, Diliman nshernandez@dcs.upd.edu.ph updilseal
Day 5
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Mathematical Logic
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1
Mathematical Induction and Recursion Sequences Recursion Mathematical Induction
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Sequences Recursion Mathematical Induction
De nition A sequence is simply an ordered list of real numbers.
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Discrete Mathematics for Computer Science
CS 30
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Sequences Recursion Mathematical Induction
Mathematical Induction and Recursion
De nition A sequence is simply an ordered list of real numbers. We express a sequence in the form {sn }n≼a and mean that our sequence consists of the terms
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sa , sa + , sa + , sa + , sa + , sa + , . . . 1
2
3
4
5
where a ∈ Z. That is, the sequence has been indexed by the integers
a, a + 1, a + 2, a + 3, a + 4, a + 5, . . . Usually, a = 0 or 1, but that need not be the case. If the indexing is clear in context, then a sequence may be displayed as {sn }, or just a formula for sn may be given.
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Sequences Recursion Mathematical Induction
Examples 1
∀n ≥ 0, sn = n!
2
∀n ≥ 1, sn = n
3
∀n ≥ 4, sn = 7 + 3n
4
∀n ≥ 0, sn = 2n
5
∀n ≥ 1, sn = 2e · 3n
6
∀n ≥ 0, sn = (−1)n (5 + 4n)
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Sequences Recursion Mathematical Induction
Examples 1
∀n ≥ 0, sn = n!
2
∀n ≥ 1, sn = n
3
∀n ≥ 4, sn = 7 + 3n
4
∀n ≥ 0, sn = 2n
5
∀n ≥ 1, sn = 2e · 3n
6
∀n ≥ 0, sn = (−1)n (5 + 4n)
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Arithmetic Sequence:
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Sequences Recursion Mathematical Induction
Examples 1
∀n ≥ 0, sn = n!
2
∀n ≥ 1, sn = n
3
∀n ≥ 4, sn = 7 + 3n
4
∀n ≥ 0, sn = 2n
5
∀n ≥ 1, sn = 2e · 3n
6
∀n ≥ 0, sn = (−1)n (5 + 4n)
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Arithmetic Sequence: ∀n ≥ 0, sn = s0 + cn
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Sequences Recursion Mathematical Induction
Examples 1
∀n ≥ 0, sn = n!
2
∀n ≥ 1, sn = n
3
∀n ≥ 4, sn = 7 + 3n
4
∀n ≥ 0, sn = 2n
5
∀n ≥ 1, sn = 2e · 3n
6
∀n ≥ 0, sn = (−1)n (5 + 4n)
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Arithmetic Sequence: ∀n ≥ 0, sn = s0 + cn Geometric Sequence: updilseal
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Sequences Recursion Mathematical Induction
Examples 1
∀n ≥ 0, sn = n!
2
∀n ≥ 1, sn = n
3
∀n ≥ 4, sn = 7 + 3n
4
∀n ≥ 0, sn = 2n
5
∀n ≥ 1, sn = 2e · 3n
6
∀n ≥ 0, sn = (−1)n (5 + 4n)
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Arithmetic Sequence: ∀n ≥ 0, sn = s0 + cn Geometric Sequence: ∀n ≥ 0, sn = s0 r n updilseal
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Sequences Recursion Mathematical Induction
Closed Formula vs. Recursive Formula A closed formula for a sequence {sn }n≼a is a formula that expresses the value of sn in terms of the index n. That is, it allows sn to be computed directly from n.
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Sequences Recursion Mathematical Induction
Closed Formula vs. Recursive Formula A closed formula for a sequence {sn }n≼a is a formula that expresses the value of sn in terms of the index n. That is, it allows sn to be computed directly from n. A recursive formula is a formula in which sn is expressed not only in terms of n but also in terms of some of the earlier values sa , sa+1 , sa+2 , sa+3 , . . . , sn−1 in the sequence. More generally, a recursive function is a function whose value for a given input is expressed in terms of its values for smaller inputs.
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Sequences Recursion Mathematical Induction
Examples 1
∀n ≥ 0, sn = n!
2
∀n ≥ 1, sn = n
3
∀n ≥ 4, sn = 7 + 3n
4
∀n ≥ 0, sn = 2n
5
∀n ≥ 1, sn = 2e · 3n
6
∀n ≥ 0, sn = (−1)n (5 + 4n)
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Arithmetic Sequence: ∀n ≥ 0, sn = s0 + cn Geometric Sequence: ∀n ≥ 0, sn = s0 r n updilseal
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Discrete Mathematics for Computer Science
CS 30
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Sequences Recursion Mathematical Induction
Mathematical Induction and Recursion
More Examples 1 2 3
s = 1, s = 2, and ∀n ≥ 2, sn = sn− + 2sn− s = 1, s = −1, and ∀n ≥ 0, sn+ = 5sn − 3sn+ s = 2, s = 1, and ∀n ≥ 2, sn = (sn− ) − (s n− ) 0
1
0
1
1
2
1
2
2
1
1
2
2
2
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Discrete Mathematics for Computer Science
CS 30
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Sequences Recursion Mathematical Induction
Mathematical Induction and Recursion
More Examples 1 2 3 4
s = 1, s = 2, and ∀n ≥ 2, sn = sn− + 2sn− s = 1, s = −1, and ∀n ≥ 0, sn+ = 5sn − 3sn+ s = 2, s = 1, and ∀n ≥ 2, sn = (sn− ) − (s n− ) F (1) = 1, F (2) = 1, and ∀n ≥ 3, F (n) = F (n − 1) + F (n − 2) 0
1
0
1
1
2
1
2
2
1
1
2
2
2
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Sequences Recursion Mathematical Induction
An Illustration
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Sequences Recursion Mathematical Induction
An Illustration
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Sequences Recursion Mathematical Induction
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Sequences Recursion Mathematical Induction
Prove the following statements: 1
The sum of the rst n positive integers is
n(n+1) 2
.
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Sequences Recursion Mathematical Induction
Prove the following statements: 1
2
The sum of the rst n positive integers is n P i =0
ri =
r n + 1 −1 r −1 ,
n(n+1) 2
.
∀ integer n ≥ 0, ∀ real numbers r except 1.
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Sequences Recursion Mathematical Induction
Prove the following statements: 1
2
The sum of the rst n positive integers is n P i =0
3
ri =
r n + 1 −1 r −1 ,
n(n+1) 2
.
∀ integer n ≥ 0, ∀ real numbers r except 1.
For all integers n ≥ 0, 22n − 1 is divisible by 3.
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Sequences Recursion Mathematical Induction
Prove the following statements: 1
2
The sum of the rst n positive integers is n P i =0
3 4
ri =
r n + 1 −1 r −1 ,
n(n+1) 2
.
∀ integer n ≥ 0, ∀ real numbers r except 1.
For all integers n ≥ 0, 22n − 1 is divisible by 3. For all integers n ≥ 3, 2n + 1 < 2n .
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updilseal
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Sequences Recursion Mathematical Induction
Prove the following statements: 1
2
The sum of the rst n positive integers is n P i =0
3 4 5
ri =
r n + 1 −1 r −1 ,
n(n+1) 2
.
∀ integer n ≥ 0, ∀ real numbers r except 1.
For all integers n ≥ 0, 22n − 1 is divisible by 3. For all integers n ≥ 3, 2n + 1 < 2n . The recursive formula: s1 = 2, ∀n ≥ 2, sn = 5sn−1 and the closed formula: sn = 2 · 5n−1 , ∀n ≥ 1 represents the same sequence.
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Sequences Recursion Mathematical Induction
Another Example Observe that 1 1−4 1−4+9 1 − 4 + 9 − 16 1 − 4 + 9 − 16 + 25
= 1 = −(1 + 2) = 1+2+3 = −(1 + 2 + 3 + 4) = 1+2+3+4+5
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Guess a general formula and prove it by mathematical induction.
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Discrete Mathematics for Computer Science
CS 30
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Sequences Recursion Mathematical Induction
Mathematical Induction and Recursion
How about this? Suppose that h0 , h1 , h2 , . . . is a sequence de ned as follows:
h = 1, h = 2, h = 3, hn = hn − + hn − + hn − for all integers n ≥ 3 0
1
1
2
2
3
Prove that hn ≤ 3n for all integers n ≥ 0.
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Sequences Recursion Mathematical Induction
The Tower of Hanoi
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Sequences Recursion Mathematical Induction
The Tower of Hanoi
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Sequences Recursion Mathematical Induction
The Tower of Hanoi
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updilseal
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Sequences Recursion Mathematical Induction
The Tower of Hanoi
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Discrete Mathematics for Computer Science
CS 30
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Mathematical Induction and Recursion
Questions?
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See you next meeting!
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Discrete Mathematics for Computer Science
CS 30
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