fskik 1 nota
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MATHEMATICAL INDUCTION
MTK3013DISCRETE STRUCTURES
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What is induction?
• A method of proof
• It does not generate answers: it only can prove them
• Three parts:– Base case(s): show it is true
for one element– Inductive hypothesis: assume
it is true for any given element• Must be clearly labeled!!!
– Show that if it true for the next highest element
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• Show that the sum of the first n odd integers is n2
– Example: If n = 5, 1+3+5+7+9 = 25 = 52
– Formally, Show
• Base case: Show that P(1) is true
Induction example
11
11)(21
1
2
i
i)1P(
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4
• Inductive hypothesis: assume true for k– Thus, we assume that P(k) is true, or that
– Note: we don’t yet know if this is true or not!
• Inductive step: show true for k+1– We want to show that:
k
i
ki1
212
1
1
2)1(12k
i
ki
Induction example, continued
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5
12121)1(2 2
1
kkikk
i
121)1(2 22 kkkk
Induction example, continued
• Recall the inductive hypothesis:
• Proof of inductive step:
k
i
ki1
212
1212 22 kkkk
21
1
)1(12
kik
i
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What did we show• Base case: P(1)• If P(k) was true, then P(k+1) is true
– i.e., P(k) → P(k+1)
• We know it’s true for P(1)• Because of P(k) → P(k+1), if it’s true for P(1), then it’s true for P(2)• Because of P(k) → P(k+1), if it’s true for P(2), then it’s true for P(3)• Because of P(k) → P(k+1), if it’s true for P(3), then it’s true for P(4)• Because of P(k) → P(k+1), if it’s true for P(4), then it’s true for P(5)• And onwards to infinity
• Thus, it is true for all possible values of n
• In other words, we showed that: )P()1P()P()1P( nnkkk
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The idea behind inductive proofs
• Show the base case• Show the inductive hypothesis• Manipulate the inductive step so that you can
substitute in part of the inductive hypothesis• Show the inductive step
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Second induction example
– Show the sum of the first n positive even integers is n2 + n
– Rephrased:
• The three parts:– Base case– Inductive hypothesis– Inductive step
n
i
nninnn1
22)P( where)P(
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Second induction example, continued
• Base case: Show P(1):
• Inductive hypothesis: Assume
• Inductive step: Show
22
11)(2)1P(1
1
2
i
i
k
i
kkik1
22)P(
)1()1(2)1P( 21
1
kkikk
i
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2323 22 kkkk
1)1()1(2 22 kkkkk
1)1(2)1(2 2
1
kkikk
i
1)1(2 21
1
kkik
i
Second induction example, continued
• Recall our inductive hypothesis:
k
i
kkik1
22)P(
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Notes on proofs by induction
• We manipulate the k+1 case to make part of it look like the k case
• We then replace that part with the other side of the k case
k
i
kkik1
22)P(
2323 22 kkkk
1)1()1(2 22 kkkkk
1)1(2)1(2 2
1
kkikk
i
1)1(2 21
1
kkik
i
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Third induction example
• Rosen, question 7: Show
• Base case: n = 1
• Inductive hypothesis: assume
6
)12)(1(
1
2
nnni
n
i
116
61
6
)12)(11(1
2
1
1
2
i
i
6
)12)(1(
1
2
kkki
k
i
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Third induction example
• Inductive step: show 6
)1)1(2)(1)1)((1(1
1
2
kkki
k
i
6
)32)(2)(1()1(
1
22
kkkik
k
i
6139261392 2323 kkkkkk
)32)(2)(1()12)(1()1(6 2 kkkkkkk
6
)32)(2)(1(
6
)12)(1()1( 2
kkkkkk
k
6
)1)1(2)(1)1)((1(1
1
2
kkki
k
i
6
)12)(1(
1
2
kkki
k
i
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Third induction again: what if your inductive hypothesis was wrong?
• Show:
• Base case: n = 1:
• But let’s continue anyway…• Inductive hypothesis: assume
6
)22)(1(
1
2
nnni
n
i
6
71
6
71
6
)22)(11(1
2
1
1
2
i
i
6
)22)(1(
1
2
kkki
k
i
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Third induction again: what if your inductive hypothesis was wrong?
• Inductive step: show 6
)2)1(2)(1)1)((1(1
1
2
kkki
k
i
6
)42)(2)(1()1(
1
22
kkkik
k
i
816102614102 2323 kkkkkk
)42)(2)(1()22)(1()1(6 2 kkkkkkk
6
)42)(2)(1(
6
)22)(1()1( 2
kkkkkk
k
6
)2)1(2)(1)1)((1(1
1
2
kkki
k
i
6
)22)(1(
1
2
kkki
k
i
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Fourth induction example
• Rosen, question 14: show that n! < nn for all n > 1
• Base case: n = 22! < 22
2 < 4
• Inductive hypothesis: assume k! < kk
• Inductive step: show that (k+1)! < (k+1)k+1
)!1( k !)1( kk kkk )1( kkk )1)(1( 1)1( kk
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