Discrete Mathematics

1
Discrete Mathematics

• Chapter 4 ??(Induction)???(Recursion)

???? ????? ???
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4.1 ?????

• ??????????

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????????????????
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(No Transcript)
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P(n) ??????? (?? n ? 2n)
?????????? P(n) ??????n??? ????????
1. ????(Basis) ??P(1)??
(?n??1?????) (? n ? 0 ????
P(0)??,??????n??????) 2. ????(Inductive)
???k???????,?P(k)???P(k1)???
(???n??k?????, ?????,????n??k1???
??)

• ?? ???????????????????,????
  ????.

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Example 1 ???n?????, .

• Proof.
• (Basis) n1? ??
• (Inductive) ??nk?????,
• ?
• ?nk1?,
• ?? 12k(k1)
• 
  ??,???????

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• Example 2. ????????n???????n2.
• ??????????????
• Pf
• Basis n1? 1 12 ??
• Inductive ??nk?????,?
• 135(2k-1) k2
• ?nk1?,?? 135(2k-1)(2k1)
• k2
  2k1 (k1)2 ??
• ???????????

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• Example 5. ???????????????n,??? n lt 2n ????
• pf
• Basis n1?,1 lt 21 ??.
• Inductive
• ??nk ?????,? k lt 2k.
• ? nk1?
• ?? k 1
• lt 2k 1
• ? 2k 1 ??
• ? k 1 lt 2k 1 ???
• ???????????

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Exercise 4.1
3. ???n???????,

• 7. ???n? ?0 ????,

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(??)
Example 7. The harmonic numbers Hk, k 1,2,3,,
are
defined by . Use
MI to show that
whenever n is a nonnegative integer.
Pf Let P(n) be the proposition that
. Basis step P(0) is true,
since .
Inductive step Assume that P(k) is true for
some k, i.e.,
Consider P(k1)
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(??)
?P(k1) is true. By MI, P(n) is true for all
n?Z.
Exercise 7, 13
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4.2 Strong Induction(??????)
(??)

• Basis step ??
• Inductive step Assume all the statements
  P(1), P(2), , P(k) are true.
  Show that P(k1) is
  also true.

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(??)
Example 2. ??? n? gt1???,? n ?????????? Pf ?
P(n) ?? n ????????? ????. Basis P(2)
is true, ?? 2 ??? Inductive ?? P(2),
P(3), , P(k) ??true. ?? P(k 1)
Case 1 k 1 is prime ?
P(k1) is true Case 2 k 1
is composite,
i.e., k 1 ab where 2 ? a ? b lt k1
????????, a ? b??????????.
? k 1??????? ? P(k1)
is true. ??????????.
Note ??????????????
13
(??)
Example 4. Prove that every amount of postage of
12 cents or more can be formed using just 4-cent
and 5-cent stamps. Pf Let P(n) be the statement
that the postage of n cents can
formed using just 4-cent and 5-cent stamps.
Basis P(12) is true, since 12 4 ? 3
P(13) is true, since 13 4 ? 2 5
? 1 P(14) is true, since 14
4 ? 1 5 ? 2 P(15) is
true, since 15 5 ? 3 Inductive Assume
P(12), P(13), , P(k) are true.
Consider P(k1) Suppose k-3
4 ? m 5 ? n.
Then k1 4 ? (m1) 5 ? n.
? P(k1) is true. By Strong MI, P(n)
is true if n?Z and n ?12.
(k-3 ? 12)
Exercise 7
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4.3 ????.

• Def. ???????????????????recursion(??).
• ? ???????? 1, 2, 4, 8, 16, 32, 64, ?
• (1) ???????,???
• an2n, n0,1,2,
• (2) ???????,???
• ??? a01,
• ??? an12an,
  n0, 1, 2,

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Example 1. ???? f ?????????
f(0)3, f(n1) 2f(n)3
?? f(1), f(2), f(3), f(4).
Sol f(1) 2f(0) 3 9
f(2) 2f(1) 3 21 f(3) 2f(2) 3
45 f(4) 2f(3) 3 93
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Example 2. ?????? F(n) n! ??????

• Sol
• ??? F(0) 1
• ??? ? F(n1) (n1)!, F(n) n!
  ? F(n1) F(n) ? (n1)

Def1, Example 5. ???(Fibonacci numbers) f0, f1,
f2, ???? ??? f0 0, f1 1,
??? fn fn-1
fn-2, for n 2,3,4, What is f5
? Sol f2 f1 f0 1
f3 f2 f1 2 f4 f3 f2 3
f5 f4 f3 5
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Exercise 4.3

• 1. ?? f(0)1 ???????? f(n) (n 0, 1, 2, ),
  ?? f(1), f(2), f(3), f(4)? (b) f(n1)
  3f(n) (d) f(n1) f(n)2 f(n) 1

3. ?? f(0) -1,f(1) 2 ???????? f(n) (n 0,
1, 2, ),?? f(2), f(3), f(4), f(5)? (a)
f(n1) f(n) 3f(n-1)
8. ????an (n 1, 2, 3, ) ?????,? (a) an
4n - 2 (c) an n (n - 1)
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Example 6. Show that fn gt a n-2 , where
(??)
Pf ( By Strong MI ) Let P(n) be the
statement fn gta n-2 . Basis f3 2 gt a

so that P(3) and
P(4) are true. Inductive Assume that
P(3), P(4), , P(n) are true. We
must show that P(n1) is true.
fn1 fn fn-1 gt a n-2 a n-3
a n-3(a 1)
? a 1 a 2
? fn1 gt a n-3 ? a 2 a n-1
We get that P(n1) is true. By Strong
MI , P(n) is true for all n ? 3
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(??)

• ?Recursively defined sets.
• Example 7. Let S be defined recursively by
• 3?S
• xy?S if x?S
  and y?S.
• Show that S is the of positive integers
  divisible by 3
• (i.e., S 3,
  6, 9, 12, 15, 18,
• Pf
• Let A be the set of all positive integers
  divisible by 3.
• We need to prove that AS.
• (i) A ? S (By MI)
• Let P(n) be the statement that
  3n?S
• 
  
• (ii) S ? A (??S???)
• (1) 3 ? A ,
• (2) if x?A,y?A, then 3x and 3y.
• ? 3(xy) ?
  xy?A
• ?S ? A

S A
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Definition 2. The set of strings over an
alphabet?? is denoted by ?. The empty
string is denoted by l, l ??, and wx??
whenever w?? and x??.eg. ? a, b, c
? l, a , b , c , aa , ab , ac , ba , bb ,
bc, abcabccba,
(??)
la
lb
lc

• Example 9. Give a recursive definition of
  ?l(w), the length of the string
  w??
• Sol
• initial value l(l)0
• recursive def l(wx)l(w)1 if
  w??, x??.

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(??)

• Exercise 13, 48, 49
• Exercise 39. When does a string belong to the
  set A of bit strings defined recursively by
• l?A
• 0x1?A if x?A.
• Sol
• Al, 01 , 0011, 000111,
• ??bit string a 0000111 ?
• a?A

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(??)

• Ackermanns function
• A(m, n) 2n
  if m 0
• 0
  if m ? 1 and n 0
• 2
  if m ? 1 and n 1
• A(m-1, A(m, n-1)) if
  m ? 1 and n ? 2
• Exercise 49 Show that A(m,2)4 whenever m ? 1
• Pf
• A(m,2) A(m-1, A(m,1)) A(m-1,2) whenever
  m ? 1.
• A(m,2) A(m-1,2) A(m-2,2) A(0,2)
  4.

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4.4 ?????

• ? ??,????????????????,??? ??????????
• ?. gcd(a, b) gcd(b mod a, a) (when
  a lt b) ?????b mod a 0,???gcd?
• Def 1. ??????????????????????????,????????

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Example 1. ???? n!??????, ?? n?
????.
Sol n! ????? ??? 0! 1
??? n! n ? (n-1)!.
???4! ? 4! 4?3!,
3! 3?2!,
2! 2?1!,
1! 1?0!,
???? 1! 1,
0! 1,
2! 2,
3! 6,
4! 24
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Algorithm 1. Procedure factorial(n
nonnegative integer) if n 0 then
factorial(n)1 else factorial(n) n ?
factorial(n-1)
??????
?????
Algorithm 1. Procedure factorial(n
nonnegative integer) if n 0 then
return(1) else return(n ? factorial(n-1))
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?Algorithm 1??4!???
factorial(4) 4 ? factorial(3)
4 ? 6 24
3 ? factorial(2)
3 ? 2 6
2 ? factorial(1)
2 ? 1 2
1 ? factorial(0) 1 ? 11
Algorithm 1. Procedure factorial(n
nonnegative integer) if n 0 then
factorial(n)1 else factorial(n) n ?
factorial(n-1)
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Example 2. ???? an,??????, ?? a???,n??????
Sol an?????
??? a01 ??? an a ? an-1.
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Exercise 4.4

• ?? ?????2???,?a2, n4???,
  ??????24??????

Algorithm 2. Procedure power(a number, n
nonnegative integer) if n 0 then
power(a, n)1 else power(a, n) a ?
power(a, n-1).
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Example 4. ??? gcd(a,b) ??????,?? altb.

• Sol

???gcd(a,b) gcd(b mod a, a)
???gcd(0,b) b
Algorithm 4. procedure gcd(a,b nonnegative
integers with altb) if a0 then gcd(a,b) b
else gcd(a,b) gcd(b mod a, a).
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Exercise 4.4

• 6. ?????4???,??gcd(8, 13)???

Algorithm 4. procedure gcd(a,b nonnegative
integers with altb) if a0 then gcd(a,b) b
else gcd(a,b) gcd(b mod a, a).
29. ????????????????n?,?? a01,a12,?an
an-1? an-2,n2, 3, ?
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(??)
Example 5. Search x in a1, a2,,an by Linear
Search
Sol
Alg. 5 procedure search (i, j, x integers)
if ai x then location i else if i
j then location 0 else search(i1,
j, x)
call search(1, n, x)
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Example 6. Search x from a1,a2,,an by binary
search (recursive version).
(??)
search x from ai, ai1, , aj

• Sol

Alg. 5 procedure binary_search (x , i , j
integers) m ?(ij) / 2? if x am
then location m else if (x lt am and i lt
m) then binary_search(x, i, m-1)
else if (x gt am and j gt m) then
binary_search(x, m1, j)
else location 0
call binary_search(x, 1, n)
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Example 1. Give the value of n!, n?Z
(??)
Sol
Note n! n ? (n-1)!
Alg. 1 (Recursive Procedure) procedure
factorial (n positive integer) if n 1 then
factorial (n) 1 else factorial (n) n ?
factorial (n-1)
Alg. (Iterative Procedure) procedure
iterative_factorial (n positive integer) x
1 for i 1 to n x i ? x
x n!
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? iterative alg. ???????? recursive alg.?
(??)

• ? Find Fibonacci numbers
• (Note f00, f11, fnfn-1fn-2 for n?2)

Alg. 7 (Recursive Fibonacci) procedure
Fibonacci (n nonnegative integer) if n 0
then Fibonacci (0) 0 else if n 1 then
Fibonacci (1) 1 else Fibonacci (n)
Fibonacci (n-1)Fibonacci (n-2)
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(??)
Alg.8 (Iterative Fibonacci) procedure
iterative_fibonacci (n nonnegative integer)
if n 0 then y 0 // y f0 else begin
x 0 y 1
// y f1 for i 1 to n-1
begin z x y
x y
y z end end y
is fn
i 1 i 2 i 3
z f2 f3 f4
x f1 f2 f3
y f2 f3 f4
Exercise 11, 35