Хаггарти р дискретная математика для программистов,...
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программирования Р. ХАГГАРТИ ТЕХНОСФЕРА Москва 2012 Допущено УМО вузов РФ по образованию в области прикладной математики в качестве учебного пособия для студентов высших учебных заведений, обучающихся по направлению подготовки «Прикладная математика» Дискретная математика для программистов Издание 2−е, исправленное Перевод с английского под редакцией С.А. Кулешова с дополнениями А.А. Ковалева, В.А. Головешкина, М.В. Ульянова
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Transcript of Хаггарти р дискретная математика для программистов,...
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2012
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2,
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519.854 22.176 1313 .
2-,
: , 2012. 400 ., ISBN 978-5-94836-303-5
, - . , , - . , , , . , - . ( ) .
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, , , -.
519.854 22.176
Pearson Education Limited 2002 This translation of DISCRETE MATHEMATICS FOR COMPUTING, First Edition is published by arrangement with Pearson Education Limited. 2012, , , ,-,
ISBN 978-5-94836-303-5ISBN 0-201-73047-2 (.)
-
................................................ 6
.................................................................. 9
1.
........................................................................ 11
1.1. ............................................................ 11
1.2. ................................................................... 14
1 ........................................................... 19
.................................................. 21
2.
............................................. 23
2.1. ................................................. 23
2.2. ................................................. 27
2.3. .................................................. 30
2.4. ............................................. 32
2 ........................................................... 35
.................................................. 38
. ................................ 39
3.
.......................................................... 44
3.1. .................................... 44
3.2. ........................................................ 51
3.3. ...................................... 53
3 ........................................................... 58
.................................................. 61
. .................................... 63
4.
.................................................................... 68
4.1. .................................................... 68
4.2. ..................................................... 73
4.3. .......... 77
4 ........................................................... 82
.................................................. 85
. .................. 86
5.
........................................................................ 91
5.1. ................ 91
5.2. ..................................................................... 96
5.3. ...................... 102
5.4. ......................................................... 105
-
4
5 ........................................................... 108
.................................................. 112
. ....... 113
6.
. ............................................................ 117
6.1. ...................................... 117
6.2. .............................................. 120
6.3. ........................................................... 128
6 ........................................................... 131
.................................................. 135
. .............................. 136
7.
............................................................................ 141
7.1. .................................................. 142
7.2. .................................................... 147
7.3. ....................................................................... 152
7 ........................................................... 158
.................................................. 163
. ......................................... 165
8.
............................................ 171
8.1. ............................................... 171
8.2. .......................................................... 175
8.3. ......................................................... 181
8 ........................................................... 184
.................................................. 187
. .................................. 189
9.
............................................................. 194
9.1. ............................................................. 194
9.2. ............................................................... 200
9.3. ................................................. 205
9 ........................................................... 208
.................................................. 211
. 2- ................ 212
................................................... 217
.................................. 275
.1. ......................................... 275
.1.1. -
........................................................... 277
-
5
.1.2.
.................................................................. 278
.1.3.
............................................. 279
.2. ................................................... 281
.2.1. , 282
.2.2. ............................. 286
.3. ........................................................ 288
.3.1. .......... 289
.3.2. ................................................... 292
.4. ........................................ 294
.4.1. .......................................... 300
............................... 305
......................................................... 305
.5. ............ 305
.............................................................. 305
.5.1. ....................... 306
.5.1 ..................................................... 317
.5.2. .................................... 318
.5.2 ..................................................... 332
.5.3. . -
................................................................ 333
.5.3 ..................................................... 344
.5.4. .. 345
.5.4 ..................................................... 359
.6.
........... 359
.............................................................. 359
.6.1. m- -
............................................................ 361
.6.2. , -
............................. 362
.6.3. NP- ................... 366
.6.4. -
............................................ 368
.6.5. -
.................... 372
.6.6. ........ 381
.6........................................................ 392
................................................................... 395
................................................ 397
-
:
= 15
P P 24
P Q P Q 25
P Q P Q 25
P Q P Q 27 28 28n! n 37
{P}A {Q} - A 39a S a | S 45a 6 S a S 45{x : P (x)} x,
P (x) 45
46
N 46
Z 46
Q 46
R 46
A S A | S 46A B A B 47A B A B 47A \B A B 48U 48
A A 48
A B A B 49
|S| S 54(a; b) 55
AB A B 55R2
56
A
n
n -
A 57
P(A) 61M(i; j) ,
i- j- 71
xR y (x; y) R 72
-
7
R
R 75
E
x
x 78
x y x | - y 80
87
88
89
R
1 91
S R R S 92MN M N 94
f(x) x 97
f : A B A B 97f(A) f 97
f
1: A 102
g f f g 104|x| x 110x x 110P (n; k) (n; k)-
121
C(n; k) (n; k)-
123
O(g(n)) , -
, g(n)
137
(v) 143
G = (V; E) V
E 143
(G) 146
K
n
n 148
154
P 160
-
171
M
k
k
M 176
M
176
d[v v 182
p p 195
p q p q 195p q p q 195{ { 200
-
8
a
b
a b 205
a a 205
a
b
a b 205
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{ { 209
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| 120 147 142 107 81
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147 132 | 108 66 105
142 42 108 | 168 61
107 157 66 168 | 112
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3 2
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168 108
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45
107
147
120
81
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1.3, 1.4 1.5 ,
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42
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45
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42
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1.4.
66
105
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1.5.
1.2.
, . -
.
begin
,
end
-
, : -
.
-
1.2. 15
:
:
=
1.2.1. ( , First Seond ,
Sum.)
begin
Input First and Seond ;
Sum
:
=First + Seond ;
end
,
.
:
; ; . ,
-
, . -
:
begin
1;
2;
.........
n;
end
1.2.2. ( : One
Two.)
begin
Input One and Two;
Temp
:
=One;
One
:
=Two;
Two
:
=Temp;
end
, , -
One Two 5 7 -
, . 1.2.
-
16 1.
1.2
Temp One Two
1 | 5 7
2 5 5 7
3 5 7 7
4 5 7 5
-
. if-then
if-then-else. :
begin
if then
end
:
begin
if then 1
else 2
end
1.2.3. ( n -
ab.)
begin
Input n;
if n < 0 then ab
:
=nelse ab
:
=n;
Output ab;
end
, ,
n, |
( ). ,
, else:
begin
Input n;
if n < 0 then n
:
=n;ab
:
=n;
Output ab;
end
-
n .
-
1.2. 17
,
.
:
for X
:
=A to Z do ; (1)
while do ; (2)
repeat
1;
2;
.............
n;
until .
(3)
X | , A Z | .
(1) . -
:
for do
(2) ,
, , , .
, .
, (3)
, . -
(2) (3) , -
,
.
1.2.4. ( n
.)
begin
sum
:
= 0;
for i
:
= 1 to n do
begin
j
:
= i i;sum
:
= sum+ j;
end
Output sum;
end
n = 4, . 1.3
-
18 1.
1.3
i j Sum
| | 0
1 1 1
2 4 5
3 9 14
4 16 30
: sum = 30.
1.2.5. (
,
.)
begin
v
:
= ;
u
:
= ;
v u;
while do
begin
u
:
= ,
;
u
;
end
end
| , -
.
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( ) (-
. 7, . 146).
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1
1.1. . 1.6 , -
.
. , -
, .
A
2
3
B
D
EF
G
3
6
3
4
3
1
2
3
5
1.6.
1.2.
() n = 3;
() n = 5.
begin
f
:
= 1;
Input n;
for i
:
= 1 to n do
f
:
= f i;Output f ;
end
-
20 1.
-
n?
1.3. i j -
m = 3 n = 4:
begin
Input m, n;
i
:
= 1;
j
:
=m;
while i 6= n dobegin
i
:
= i+ 1;
j
:
= j m;end
Output j;
end
m n > 0. n = 0?
1.4. ,
:
begin
first
:
= 1;
Output first;
seond
:
= 1;
Output seond;
next
:
= first+ seond;
while next < 100 do
begin
Output next;
first
:
= seond;
seond
:
=next;
next
:
= first+ seond;
end
end
.
1.5. l, sum k -
, n = 6.
-
21
begin
Input n;
k
:
= 1;
l
:
= 0;
sum
:
= 0;
while k < 2n do
begin
l
:
= l + k;
sum
:
= sum+ l;
k
:
= k + 2;
end
Output sum;
end
-
n.
1.6. ,
. 1.1. ?
begin
: 1, 2, 3, . . . . .;
m
:
= ;
:
= ;
:
= 1;
while > m 1 dobegin
if
then
begin
;
:
= 1;end;
:
=+ 1;
end
end
-
,
.
-
22 1.
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-
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, . . ( -
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1
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.
-
24 2.
. ,
, P , Q | -
| R | 29 | .
, , , ,
, , -
. ,
( P ) | ; (P Q) | | ; (P Q) | | .
2.1. P | -
, Q| .
.
() | , .
() , | .
() | , .
.
() ( P ) Q.
() ( P ) ( Q).
() P Q.
-
, -
, . .
.
.
P -
( P ), -
P .
. 2.1.
2.1
P ( P )
-
2.1. 25
P Q (P Q). -
,
. -
.
| . 2.2.
2.2
P Q (P Q)
P
Q (P Q). ,
,
-
. , (P Q) , P ,
Q, , . -
. 2.3.
2.3
P Q (P Q)
2.2. -
: VIII
, ,
1
?
. P -
, Q | VIII
R| . -
: (P Q) ( R). , P ,
Q R . (P Q) ( R)
: ( ) , .
1
| , -
XVII{XVIII .. -
. | . .
-
26 2.
,
, , -
. -
.
2.3. , ( (P ( Q))) -
(( P ) Q).
. (. 2.4)
:
R = ( (P ( Q))) S = (( P ) Q):
-
P Q.
2.4
P Q P Q P ( Q) R S
. , -
R S.
, -
. -
: ,
| .
, -
.
P , Q. ,
P , -
, Q . P ,
Q , -
.
2.4. P | () 1 = 5, Q |
( ) 3 = 7 R | () -
4 = 4. , : P , Q
P , R, | .
. 1 = 5, , 2 ,
, 3 = 7. , P , Q
-
2.2. 27
. 1 = 5
3 2 = 2. (2)2 = 22, . . 4 = 4. , P , R .
P , Q -
, P ,
Q . -
.
, P Q - P , Q. -
P Q , P Q, P
Q, Q P .
. 2.5.
2.5
P Q (P Q)
2.5. (( Q) ( P )) - (P Q)., (( Q) ( P )) - (P Q).. (. 2.6).
2.6
P Q P Q (P Q) (( Q) ( P ))
,
, , .
2.2.
-
, | ,
. , ,
-
28 2.
.
, ,
. , x | , -
x = x
2
,
x = 0 x = 1 .
. -
-
. -
, (-
),
.
2.6. ,
?
() 180
.
() .
() x, x
2
= 2.
() .
.
() .
() .
1
.
() .
() . 2 , .
2.6 -
, -
, ()
, .
() -
, , . , .
.
1
| . | . .
-
2.2. 29
2.7. P (x) x |
x
2
= 16. : x : P (x) .
. x : P (x) , - x, x
2
= 16. ,
, , x
2
= 16 -
x = 4. , x = 4 | . -
x, x : P (x).
2.8. P (x) | : x |
x
2
+1 = 0. : x : P (x) .
. :
x, x
2
+ 1 = 0. -
, . .
x
2 > 0, , x2 + 1 > 1. ,
x : P (x) . 2.8 -
: x : P (x). , , , , x, -
x
2
+ 1 = 0. ,
x, x
2
+ 1 6= 0. x P (x).
P (x) -
2
:
x : P (x) x P (x); x P (x) x : P (x).
, -
, .
2.9. , x y | ,
P (x; y) x+ y = 0. -
.
() x y : P (x; y);() y : x P (x; y).
2
. | . .
-
30 2.
.
() x y : P (x; y) , x y,
x+y = 0. , , , x
, y = x x + y = 0 .
() y : x P (x; y) : y, -
x x+ y = 0. , -
, : y,
. , .
2.3.
.
|
. -
,
(P Q). , :
1. . , P -
Q. -
, P , Q | , -
(P Q) (. . 2.5 . 27).
2. . , Q
P . , ,
(( Q) ( P )), 2.5, -
(P Q).3. . , P
, Q , , -
. - ,
(P Q) , - P , Q .
2.10. , -
xy x y .
-
2.3. 31
. , ,
x, x = 2m + 1, m |
. , y = 2n+ 1 n.
,
xy = (2m+ 1)(2n+ 1) = 4mn+ 2m+ 2n+ 1 = 2(2mn+m+ n) + 1
.
2.11. n | . , -
, n
2
, n
.
. n
2
-
n
2
, n -
n . ,
,
n n
2
.
n , n = 2m - m. -
, n
2
= 4m
2
= 2(2m
2
) | .
2.12. ,
x
2
= 2 , . .
.
. , x
x
2
= 2 , . . x =
m
n
m n, n 6= 0. , , - -
.
,
. , x =
m
n
=
2m
2n
=
3m
3n
. . -
, m n .
.
, , x =
m
n
-
(m n ). x -
x
2
= 2. ,
(m
n
)2
= 2, m
2
= 2n
2
.
, m
2
. -
, m (. . ())
m = 2p - p. -
m
2
= 2n
2
, , 4p
2
= 2n
2
, . . n
2
= 2p
2
.
n . , -
, m, n| .
-
32 2.
2. , -
m
n
,
.
:
x
2
= 2 ,
. . .
2.4.
, , -
. ,
- -
, , -
. -
, .
, , -
, -
.
, -
, -
a
1
; a
2
; a
3
; : : : ; a
n
.
begin
i
:
= 0;
M
:
= 0;
while i < n do
begin
j
:
= j + 1;
M
:
=max(M; a);
end
end
: a
1
= 4, a
2
= 7, a
3
= 3
a
4
= 8 . 2.7.
2.7
j M j < 4?
0 0
1 4
2 7
3 7
4 8
-
2.4. 33
M = 8, -
. , -
M ,
.
n?
a
1
; a
2
; a
3
; : : : ; a
n
n -
M
k
M k- .
1. a
1
1,
M 0 a
1
,
, , a
1
( 0).
.
2. k- M
k
|
a
1
; a
2
; : : : ; a
k
, M
k+1
-
max(M
k
; a
k+1
), . .
a
1
; a
2
; : : : ; a
k
; a
k+1
.
. 1 , -
1. . 2, -
2. . 2
, ,
3, . . , -
n, . .
.
-
.
P (n) | , -
n.
,
1. P (1)
2. k > 1 (P (k) P (k + 1)) . P (n) n.
2.13. ,
1 + 2 + + n = n(n+ 1)2
n.
-
34 2.
. P (n) | 1 + 2 + + n = n(n+1)2
.
n = 1 | 1,
,
1(1 + 1)
2
= 1:
, P (1) .
, 1+2+ +k = k(k+1)2
- k.
1 + 2 + + k + (k + 1) = (1 + 2 + + k) + (k + 1) =
=
k(k + 1)
2
+ (k + 1) =
=
1
2
(k(k + 1) + 2(k + 1)
)=
=
1
2
((k + 2)(k + 1)
)=
=
(k + 1)(k + 2)
2
:
, k
P (k) P (k + 1)
. , , -
P (n) n.
2.14. ,
7
n 1 6 n.. , a
b ,
a = mb - m. , 51 17,
51 = 3 17. , - , ,
b b.
P (n) 7
n 1 6. n = 1
7
n 1 = 7 1 = 6;. . P (1) .
, 7
k 1 6 - - k.
-
2 35
7
k+1 1 = 7(7k) 1 == 7(7
k 1) + 7 1 == 7(7
k 1) + 6: 7
k1 6, 7(7
k 1) + 6 6., 7
k+1 1 6, k (P (k) P (k + 1)) .
P (n) n.
2.15. x
1
; x
2
; : : : ; x
n
-
:
x
1
= 1 x
k+1
= x
k
+ 8k k > 1:
, : x
n
= (2n 1)2 n > 1.. x
n
= (2n 1)2 P (n). n = 1, (2n 1)2 = (2 1)2 = 1, P (1).
, x
k
= (2k1)2 k > 1.
x
k+1
= x
k
+ 8k =
= (2k 1)2 + 8k == 4k
2
+ 4k + 1 =
= (2k + 1)
2
:
, x
k+1
=
(2(k+1) 1)2 -
(P (k) P (k + 1)) k > 1. , , P (n)
-
n.
2
2.1. P , Q R | -
:
P : .
-
36 2.
Q: .
R: .
-
, P , Q R.
() .
() , .
() , .
() , .
() , .
2.2. P : , Q |
. -
:
() , ;
() ;
() , ( -
)
.
, -
() ().
2.3. ,
, -
. -
:
() (P ( P ));
() P ( P );() (P (P Q)) Q.
2.4. , (P Q) R - (( P ) R) (Q R).
2.5. x , P (x) x
.
:
() ;
() ;
() .
-
2 37
() ,
() ,
.
2.6. P (x) x , Q(x) | x ,
x | - . :
x (P (x) Q(x)):
:
() ;
() ;
() .
2.7. () -
:
n m | n+m | .() :
n
2
| n | .() ,
n+m | , | .
2.8.
.
() 1+5+9+ +(4n3) = n(2n1) n.
() 1
2
+2
2
+ +n2 = 16
n(n+1)(2n+1)
n.
()
1
13 +1
35 + + 1(2n1)(2n+1) =
n
2n+1
n.
() n
3 n 3 - n.
() 1 1!+2 2!+ +n n! = (n+1)!1 n.
( n! n -
1 n :
n! = 1 2 3 (n 1) n.)
-
38 2.
2.9. x
1
; x
2
; : : : ; x
n
-
x
1
= 1 x
k+1
=
x
k
x
k
+ 2
k > 1:
x
2
, x
3
x
4
. ,
x
n
=
1
2
n 1 n > 1.
2.10. x
1
; x
2
; : : : ; x
n
-
x
1
= 1; x
2
= 2 x
k+1
= 2x
k
xk1 k > 1:
x
3
, x
4
x
5
. x
n
.
-
.
,
, . . .
-
. -
, , ... .
. 2.8 ,
. 2.8
P Q P Q P Q (P Q)
-
,
.
-
. 39
x
, , : P (x).
() () | .
(P Q) P Q.
( Q P ) (P Q).
(P Q), - Q P ,
.
-
, .
| :
P (n) | , -
n.
,
1. P (1)
2. k > 1 (P (k) P (k + 1)) . P (n) n.
.
( , -
, , ), -
, , -
.
.
P | ,
A, Q| , , -
. {P}A {Q} , A -
P , Q. P
, Q |
-
40 2.
. {P}A {Q} - .
A {P}A {Q}. .
1. .
begin
z
:
=x y;end
. P :
x = x
1
y = y
1
. Q | z = x
1
y1
.
{P} {Q}
x = x
1
y = y
1
, z = x
1
y1
. -
x = x
1
y = y
1
, z, x y. -
: z = x y, x = x1
y = y
1
z = x
1
y1
.
A
, A
1
; : : : ; A
n
{P}A1
{Q1
}; {Q1
}A2
{Q2
}; : : : ; {Qn1}An {Q};
.
2. -
.
{x | }begin
y
:
= ax;
y
:
= (y + b)x;
y
:
= y + ;
end
{y = ax2 + bx+ }
. ,
- .
-
. 41
P {x = x1
}begin
y
:
= ax;
Q
1
{y = ax1
x = x
1
}y
:
= (y + b)x;
Q
2
{y = ax21
+ bx
1
}y
:
= y + ;
end
Q {y = ax21
+ bx
1
+ }
, , , :
{P} y := ax {Q1
},{Q
1
} y := (y+ b)x {Q2
},{Q
2
} y := y+ {Q}, |
. ,
{P} {Q}
, . . .
.
if ... then,
.
: , -
if then
1;
else
2;
P , Q.
:
{P } 1 {Q}
{P ()} 2 {Q}.
-
42 2.
3. , .
{x | }begin
if x > 0 then
ab
:
=x;
else
abs
:
=x;end
{abs | x}
. P {x = x1
}, Q {abs| x}.
{P x > 0} abs := x {Q} , x
1
-
.
{P (x > 0)} abs := x {Q} , x
1
.
- ,
while ... do, -
.
.
4. -
.
{n | }begin
sq
:
= 0;
for i
:
= 1 to n do
sq
:
= sq + 2i 1;end
{sq = n2}
. P (n) sq = n
2
n-
, sq
k
| sq k-
. ,
(1) sq
1
= 1
2
;
(2) sq
k
= k
2
, sq
k+1
= (k + 1)
2
.
-
. 43
, sq
1
= 1 (1)
. , k- sq
k
= k
2
.
sq
k+1
= sq
k
+ 2(k + 1) 1 = k2 + 2k + 1 = (k + 1)2:
, (2) .
, , P (1) (. (1)). ,
((P (k) P (k + 1)) k > 1. ,
, P (n) n.
4 for
(). , -
, while ... do, -
, -
.
, .
-
3
| ,
, .
-
-
. , ,
.
, -
,
. -
, ,
, , -
. ,
, , , -
, (
).
.
, I.
3.1.
| , -
. ,
{, , }; {2, 3, 5, 7, 11}; {, , , }.
-
. , -
. -
-
3.1. 45
, S = {3; 2; 11; 5; 7} | , -. , S ,
, , -
, .
a S , a | S. , a S.
a S, : a 6 S. , -
.
.
S = {x : P (x)} , x,
P (x) . ,
S = {x : x | }
S = {1; 3; 5; 7; : : :}:
2n 1, n | , :
S = {2n 1 : n | }:
3.1. , -
.
() A = {x : x | x2 + 4x = 12};() B = {x : x | , };() = {n2 : n | }.
.
() x
2
+4x = 12, x(x+4) = 12. x| ,
12, 1, 2, 3, 4,6 12. , x + 4 12. : x = 6 x = 2. -
x
2
+ 4x 12 = 0. x = 6 x = 2., A = {6; 2}.
-
46 3.
() B = {, , }.() = {0; 1; 4; 9; 16; : : :}.
, -
.
| ;
N = {1; 2; 3; : : :} | ;Z = {0; 1; 2; 3; : : :} | ;Q = {p
q
: p; q Z; q 6= 0} | ;
R = { }| . , -
N 0.
, -
.
. -
,
.
-
. .
, -
-
. , = {0; 1; 4; 9; 16; : : :} Z = {0; 1; 2; 3; : : :}.
, A -
S,
S. ,
A S. : A S. . 3.1 . -
.
, -
.
, .
A = B
:
{x A x B} {x B x A}:
-
3.1. 47
S
A
3.1. A S
3.2.
A = {n : n2 | }
B = {n : n | }:, A = B.
. x A, x2 | . 2.11, , x |
. , x B, . . A B. , x B. x |
. 2.10, x
2
-
, , x A. x B , B - A, . . B A. , A = B.
A B
A B = {x : x A x B}:
, -
A, B, .
. 3.2.
A B
A B = {x : x A x B}:
, A,
B. . 3.3.
-
48 3.
A B
3.2. A B
A B
3.3. A B
1
B A
A \B = {x : x A x 6 B}:
A \B A, B (. . 3.4).
-
U , U
. -
.
A U -
A U , . . U \ A. , -
U \ A A 1
. |. .
-
3.1. 49
A. , , -
U ,
A = {x : (x A)} A = {x : x 6 A}:
. 3.5.
A B
3.4. A \ B
A
3.5. A
A B
A B = {x : (x A x 6 B) (x B x 6 A)}:
, A B,
, B, A. ,
, A, B,
. , ,
. 3.6.
-
50 3.
A B
3.6. A B
3.3.
A = {1; 3; 5; 7}; B = {2; 4; 6; 8}; C = {1; 2; 3; 4; 5}: A C, B C, A \ C B C..
A C = {1; 3; 5; 7; 2; 4};B C = {2; 4};A \ C = {7};B C = (B \C) (C \B) = {6; 8}{1; 3; 5} = {6; 8; 1; 3; 5}.
3.4.
A = {x : 1 6 x 6 12 x };B = {x : 1 6 x 6 12 x , 3}:
, (A B) = A B.. ,
U = {1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12}: ,
A = {2; 4; 6; 8; 10; 12} B = {3; 6; 9; 12}:
-
3.2. 51
(A B) = {6; 12} = {1; 2; 3; 4; 5; 7; 8; 9; 10; 11}
A B = {1; 3; 5; 7; 9; 11} {1; 2; 4; 5; 7; 8; 10; 11} == {1; 2; 3; 4; 5; 7; 8; 9; 10; 11}:
, (A B) = A B.
3.2.
, , -
.
, | , -
.
1
-
, . 3.1.
3.1
3.5. , A B
: (A B) = A B..
(A B) = {x : x 6 (A B)} == {x : (x (A B))} == {x : ((x A) (x B))};
A B = {x : (x 6 A) (x 6 B)} == {x : ( (x A)) ( (x B))}:
,
:
(P Q) ( P ) ( Q);
1
, . | . .
-
52 3.
P Q | . -
-
(. 3.1), , (P Q) -
(A B), ( P ) ( Q) | AB., (A B) = A B.
, 3.5, , -
. ,
, .
3.2.
, , -
3.5.
(. 3.2)
,
, U . - , |
.
3.2.
A (B C) = (A B) C A (B C) = (A B) C
A B = B A A B = B A
A = A A U = A
A U = U A =
A A = A A A = A
A (B C) = (A B) (A C) A (B C) = (A B) (A C)
A A = U A A =
U = = U
A = A A = A
(A B) = A B (A B) = A B
-
.
-
3.3. 53
. , ,
, - -
, .
3.6. , ,
A B :
A B = (A B) (A B):
. -
:
A B = (A B) (B A): , .
(A B) (A B) = (. )= (A B) (A B)) = (. .)= ((A B) A) ((A B) B) = (. .)= (A (A B)) (B (A B)) = (. .)= ((A A) (A B)) ((B A) (B B)) = (. .)
= ((A A) (B A)) ((A B) (B B)) = (. )= ( (B A)) ((A B) ) = (. .
)
= (A B) (B A)
,
A B = (A B) (A B); .
3.3.
6 , ,
. ,
. ,
?
?
-
54 3.
S -
. |S|. -
. ,
.
.
|A B| = |A|+ |B| |A B|:. . 3.7, A B - : A\B, AB B\A, . ,
A = (A \B) (A B) B = (B \ A) (A B):
:
|A \B| = m; |A B| = n; |B \ A| = p:
|A| = m+ n, |B| = n+ p
|A B| = m+ n+ p == (m+ n) + (n+ p) n == |A|+ |B|+ |A B|:
AB BAAB
3.7.
-
3.3. 55
3.7. 63 ,
,
. 16 , 37 |
, 5 ,
-
?
. .
A = {, };B = {, }:
|A| = 16; |B| = 37; |A B| = 5:,
|A B| = 16 + 37 5 = 48:, 6348 = 15 .
,
,
.
.
, , -
, . -
.
, -
.
(a; b), a| -
A, b | B. -
A B AB., A B = {(a; b) : a A b B}.
, -
-
,
.
3.8. A = {x; y} B = {1; 2; 3}. : AB, B A B B.. AB
{(x; 1); (x; 2); (x; 3); (y; 1); (y; 2); (y; 3)}:
-
56 3.
B A |
{(1; x); (2; x); (3; x); (1; y); (2; y); (3; y)}:
, AB B A ! - B B
{(1; 1); (1; 2); (1; 3); (2; 1); (2; 2); (2; 3); (3; 1); (3; 2); (3; 3)}:
3.8, , -
A B
1
|AB| = mn; |A| = m |B| = n:
,
.
, -
. , A B 3.8 . 3.8
x
y
( 1)x, ( 2)x, ( 3)x,
( 1)y, ( 2)y, ( 3)y,
1 2 3
B
A
3.8.
-
R . RR R
2
, ,
(x; y).
. R2
-
. . 3.9
1
, , A
( m ) n . |
. .
-
3.3. 57
A
1
,
A
2
, . . . , A
n
A
1
A2
An
= {(a1
; a
2
; : : : ; a
n
) : a
i
Ai
; i = 1; 2; : : : ; n}: | -
, , -
. -
.
.
y
x
(3,2)
3.9.
, A
1
, A
2
, . . . , A
n
-
A, A
n
n A.
3.9. B = {0; 1}. Bn.. B
n. -
n.
,
. S = {s1
; s
2
; : : : ; s
n
}, - -
. A S, n- (b
1
; b
2
; : : : ; b
n
), b
i
= 1, s
i
A bi
= 0 -
. -
A.
, -
, 1
, 0 .
-
58 3.
3.10. S = {1; 2; 3; 4; 5}, A = {1; 3; 5} B = {3; 4}. A B,
A B, A B B.. ,
A a = (1; 0; 1; 0; 1), -
B b = (0; 0; 1; 1; 0). ,
a b = (1; 0; 1; 0; 1) (0; 0; 1; 1; 0) = (1; 0; 1; 1; 1);
a b = (1; 0; 1; 0; 1) (0; 0; 1; 1; 0) = (0; 0; 1; 0; 0);
b = (0; 0; 1; 1; 0) = (1; 1; 0; 0; 1):
: AB = {1; 3; 4; 5}, AB = {3} B = {1; 2; 5}.
3
3.1. () :
A = {x : x Z 10 6 x 6 17};B = {x : x Z x2 < 24};C = {x : x Z 6x2 + x 1 = 0};D = {x : x R 6x2 + x 1 = 0}:
: 6x
2
+ x 1 = (3x 1)(2x + 1):() -
:
S = {2; 5; 8; 11; : : :};T = {1; 1
3
;
1
7
;
1
15
; : : :}.
3.2. -
U = {p; q; r; s; t; u; v; w}. A = {p; q; r; s}, B == {r; t; v} C = {p; s; t; u}. :
() B C; () A C; () C;() A B C; () (A B); () B C.() (A B) (A C); () B \ C;
-
3 59
3.3.
.
A = {x : x | , };B = {x : x | , };C = {x : x | , }.
, :
() C A B;() B C;() B C;() A B = . :
() A B C;() (A B) C.
3.4. :
A = {3n : n Z n > 4};B = {2n : n Z};C = {n : n Z n2 6 100}.
() , -
A, B C:
(i) ;
(ii) {10; 8; 6; 4; 2; 0; 2; 4; 6; 8; 10};(iii) {6n : n Z n > 2};(iv) {9; 7; 5; 3; 1; 1; 3; 5; 7; 9}.
() A \B .3.5. ,
:
A (B C) = (A B) (A C):
,
A, B C.
-
60 3.
3.6. , -
:
A (B C) = (A B) (A C):
, A (B C) - (A B) (A C).
3.7.
:
() (A B) B = A B;() (A (B C)) = A B C;() (A B C) (A B C) (A C) = ;() (A \B) \ C = A \ (B C);() A A A = A.
3.8. :
A B = (A B):
A B. - :
() A A = A;() (A A) (B B) = A B;() (A B C) (A B C) (A C) = ;() (A B) (A B) = A B.
3.9. () , -
A, B C :
|A B C| ==|A|+ |B|+ |C| |A B| |B C| |A C|+ |A B C|:
() , -
, -
. 25 -
, 27 , 12 -
. , 20 ,
,
, | . ,
-
61
-
.
?
?
3.10. A B,
AB = B A? A, B C
A B = A C. , B = C? .
3.11. A, B C | . ,
() A (B C) = (AB) (A C);() (A B) C = (A C) (B C).
3.12. P(A) , A.
, P(A) = {C : C A}.() P(A), A = {1; 2; 3}.() , P(A)P(B) = P(AB) -
A B.
() , P(A)P(B) - P(A B).
3.13. U = {1; 2; 3; 4; 5; 6} | . - :
A = {1; 2; 4; 5} B = {3; 5}:
AB A B, .
| , -
.
, U | .
N = {1; 2; 3; : : :} | .Z = {0; 1; 2; 3; : : :} | .Q = {p
q
: p; q , q 6= 0} | .R = { } | .
-
62 3.
S A, -
S. : A S. ,
.
A B
A B = {x : x A x B}: A B
A B = {x : x A x B}: B A
A \B = {x : x A x 6 B}: A ( U) -
A = {x : x 6 A}: A B
A B = {x : (x A x 6 B) (x B x 6 A)}: ,
, U . S -
. |S|. ,
|A B| = |A|+ |B| |A B|: A B -
AB = {(a; b) : a A b B}: AB . R R R2 . ( n) B
n
,
B = {0; 1}.
-
. 63
.
-
.
.
.
-
, I. -
, .
( I, II) (, I)
( III, IV) (, II)
( III, IV) (, III)
( III, ) (, IV)
(, ) (, IV)
(, VII) (, )
(VII, V) (, VII)
(V, VIII) ( , V)
(V, VI) (, VI)
(VI, II) ( II, )
(, )
(, )
( , {)
({, )
, (x; y) , x -
y, (x; y) , x | y.
, , -
, , , PROLOG.
,
. , : I
III?, , -
( I, 3) .
: ? | . -
, -
. , ? | (x, IV)
IV?.
, , x , -
, .
-
64 3.
1. :
() ? | ( II, );
() ? | (, II);
() ? | ();
() ? | (, II).
.
, -
. , -
,
.
-
, .
,
. -
, ,
, .
,
x, (x; y), . -
(1) , ,
x | y, x | .
(1) (x) from (x; y).
, (2), .
, x | y, y | x.
(2) (y; x) from (x; y).
2. 1?
:
() ? | ({);
() ? | (, );
() ? | ().
. () 1 -
, (1), :
(, IV).
() , {-
- .
-
. 65
() | , -
(, ) (2).
(),
.
, -
, (1):
(3) (y) from (x; y)
,
:
(3) (x; y) from ((x) (y; x))
3. :
() ? | ( IV);
() ? | ( IV, III);
() ? | ( IV, );
() ? | (VIII, V).
.
() (,
IV) (3).
() ( III,
IV) ()
(4).
| .
, () ()
, -
, . -
,
. , III, -
IV | , -
III. , (4)
().
() , -
VIII. (3)
? | (VIII).
-
66 3.
, , -
,
, ,
.
. , -
-
, .
, , ,
(A) (B):
(A) (x) from (x; y);
(B) (x) from ( (x)).
: ? | (VIII), -
, -
(A) (B). ? | (VIII) -
. -
(VIII) . -
(B) ? | (VIII) -
! , -
,
.
4. -
-
. (x) , -
, x |
- x | - . -
(1)
.
?
. :
(x) from ([(x; y) (z; y)
[(x) (x; y)).
[(x; y) (z; y)
: , , , -
. [(x) (x; y)
.
-
. 67
-
, . , ,
II.
.
, , -
. ,
.
-
4
, , -
, , .
(, ) -
,
(, , . . ). -
A B A B , -
,
.
S -
K .
S K - (s; k), : s
k. ... -
...,
.
-
.
. , -
: -
. -
, . -
.
n-
.
4.1.
A B -
R A B. , A = B, R A.
4.1. ,
. 4.1. ,
P :
-
4.1. 69
() R = {(x; y) : x | y};() S = {(x; y) : x | y}.
&
&
&
4.1.
.
() R : (, ), (, -
), (, ), (, ), (, ) (,
).
() S : (, ), (, ), (, -
), (, ), (, ), (, ), (, ),
(, ) (, ).
4.2. , -
A = {1; 3; 5; 7} B = {2; 4; 6}:() U = {(x; y) : x+ y = 9};() V = {(x; y) : x < y}.
.
() U : (3; 6), (5; 4) (7; 2);
() V = {(1; 2); (1; 4); (1; 6); (3; 4); (3; 6); (5; 6)}.
4.3.
R = {(x; y) : x | y}
A = {1; 2; 3; 4; 5; 6}. , .
-
70 4.
. R : (1; 1), (1; 1), (1; 2), (1; 3), (1; 4), (1; 5),
(1; 6), (2; 2), (2; 4), (2; 6), (3; 3), (3; 6), (4; 4), (5; 5) (6; 6).
-
, -
. ,
.
A B | R | -
. -
. R
, , -
.
, , ,
.
V -
A = {1; 3; 5; 7} B = {2; 4; 6} 4.2 (). - . 4.2.
1
3
5
7
2
4
6
4.2. V A B
A -
,
A, , , ,
.
4.4. , R
4.3.
. R |
A = {1; 2; 3; 4; 5; 6};
.
. 4.3.
-
4.1. 71
4.3. R A
-
. ,
R A
B.
- . :
A = {a1
; a
2
; : : : ; a
n
}; B = {b1
; b
2
; : : : ; b
m
}:
R M n
m . A,
| B ,
. ,
i- j-
M(i; j), :
M(i; j) = ; (a
i
; b
j
) R;M(i; j) = ; (a
i
; b
j
) 6 R;
nm . , U 4.2()
:
2 4 6
1
3
5
7
:
-
72 4.
, -
.
.
4.5. R A = {a; b; ; d} :
;
-
A. ,
R.
. R : (a; b),
(a; ), (b; ), (b; d), (; b), (d; a), (d; b) (d; d).
4.6. , R
4.3.
. R :
1 2 3 4 5 6
1
2
3
4
5
6
:
R| , (x; y) R - xR y. , x|
y . 4.3
x | y -
.
, -
,
:
( ); ; ; .
-
4.2. 73
4.7. R A = {1; 2; 3; 4} - . 4.7.
-
,
R, -
.
4.3.
. R = {(2; 1); (3; 2); (4; 3)}. (
) :
1 2 3 4
1
2
3
4
:
x y = 1:
4.2.
, -
.
, R A
, x A xRx;, xR y y Rx x y A;, (xR y y Rx x = y) x y A;, (xR y y R z xR z) -
x; y; z A. -
. R , (x; x) R x; ,
(x; y) R , (y; x) R; , : (x; y) R x 6= y , (y; x) 6 R; -, (x; y) R (y; z) R (x; z) R.
-
74 4.
, -
, , . . , -
. -
x
y , : y
x. ,
x y, y x -
. , ,
, x y
y z x z.
, . -
, A -
, . .
. , M , ,
, , -
(M(i; i)), ; M -
, . . M(i; j) =M(j; i); -
:(M(i; j) = i 6= j)M(j; i) = :
, -
.
4.8. (, -
, )
:
() x y ;
() x 6= y ;() x y
.
.
() x , -
. , , , , , 2
6, : 6 2. ,
. , x
y, y z. -
, y = mx
-
4.2. 75
m, | z = ny, n | -
. , z = ny = (nm)x, . . x z. ,
. , -
, : x y y
x , y = x.
() x 6= x , -. , x 6= y , y 6= x. , , , 2 6= 3 3 6= 2, , , 2 = 2. ,
x 6= y y 6= x , x = y.() , -
x .
, x
y -
y x. -
, , x, y z,
x y,
y z, -
. ,
.
R A
, R
,
. -
-
R AA , R . , R
R
. ,
R
R -
, , R
R
.
, R
R -
P ,
1. R
P ;
2. R R;3. R
, -
R P .
-
76 4.
4.9. A = {1; 2; 3}, R A - :
R = {(1; 1); (1; 2); (1; 3); (3; 1); (2; 3)}: , .
.
. -
(x; x). ,
:
R
= {(1; 1); (1; 2); (1; 3); (3; 1); (2; 3); (2; 2); (3; 3)};
.
, . ,
R
= {(1; 1); (1; 2); (1; 3); (3; 1); (2; 3); (2; 1); (3; 2)}:
, -
. R (3; 1)
(1; 2), (3; 2). -
, (2; 3) (3; 1) (2; 1), (3; 1)
(1; 3) | (3; 3). :
R
{(1; 1); (1; 2); (1; 3); (3; 1); (2; 3); (3; 2); (2; 1); (3; 3)}: (2; 1) (1; 2). , -
R
(2; 2). ,
( ,
A
2
). ,
R
= {(1; 1); (1; 2); (1; 3); (3; 1); (2; 3); (3; 2); (2; 1); (3; 3); (2; 2)}:
, -
4.9, . 8 -
, ,
-
.
. -
, , -
.
,
.
-
4.3. 77
4.3.
.
, -
A . -
-
. (. .
), , - -
.
.
... , ... . , -
,
.
R, : xR y, xy > 0 -
.
.
... , ... .
.
, -
,
. -
. |
.
A -
A
1
; A
2
, . . .A
n
A,
:
1) A = A
1
A2
An
;
2) A
i
Aj
= i 6= j. A
i
.
A -
. 4.4. , ,
-
78 4.
. ,
.
4.4.
, R -
A .
.
. E
x
-
x A Ex
= {z A : z Rx}. .
. R |
A.
A.
. .
, -
A. , E
x
|
A. , R | , . . xRx. -
, x Ex
E
x
.
, xR y E
x
= E
y
. -
, xR y z Ex
. z Rx
xR y. R | , ,
z R y. , z Ey
. , E
x
Ey
.
, E
y
Ex
, E
x
= E
y
, .
, -
, , A -
-
4.3. 79
.
, E
x
| A
-
A. , x A, x Ex
. -
, x . ,
A . ,
A .
, , , -
. , -
. -
. , E
x
Ey
6= . z A, E
x
Ey
. ,
z Rx z R y. R | , -
, xR z z R y. R, xR y.
, , E
x
= E
y
. , -
, E
x
E
y
, . -
.
.
, ,
A, -
: , -
.
4.10. R R
: xR y, x y | . , R| -
, 0,
1
2
2.
. x x = 0 Z x, R . x y , - y x = (x y) . ,R | . x y y z | . xz = (xy)+(yz) | , . . . , R .
, , R , -
. , R | .
E
x
x :
E
x
= {z R : z x | }:
-
80 4.
,
E
0
= Z;
E
1
2
= {z R : z 12
| } =
= {: : : ; 112
; 12
;
1
2
; 1
1
2
; 2
1
2
; : : :};
E
2
= {z R : z 2 | } =
= {: : : ; 1 +2;
2; 1 +
2; 2 +
2; : : :}:
, , R
A . -
, - -
. ,
, .
.
6 ; ; ... ... .
-
.
R | A,
x 6= y xR y x , y | .
y . -
x y, z,
xR z z R y, x -
1
y x y.
, . -
A,
x y, x y .
-
,
.
1
, y x. | . .
-
4.3. 81
4.11. , ... ... -
A = {1; 2; 3; 6; 12; 18}. ,
.
. .
4.1
1
2 1 1
3 1 1
6 1, 2, 3 2, 3
12 1, 2, 3, 6 6
18 1, 2, 3, 6 6
12 18
6
2 3
1
4.5.
A -
,
.
.
6 ; . ,
-
. . -
, ,
2
-
2
, . , -
Z 6 ,
. | . .
-
82 4.
( ) (
).
4.11
, , 1. -
, : 12 18. -
.
. ,
{1; 2; 6; 18} - ... ....
4
4.1. -
, :
a b d
1
2
3
:
4.2. -
N , -
:
R ={(x; y) : 2x+ y = 9};S ={(x; y) : x+ y < 7};T ={(x; y) : y = x2}:
4.3. R | {1; 2; 3; 4}, - : uR v , u + 2v |
. R :
() ;
() ;
() .
4.4. ,
, :
() ... , ...;
() ... ...;
() ... , ...;
() ... , ....
-
4 83
4.5. , Z
, , -
?
() x+ y | ;
() x+ y | ;
() xy | ;
() x+ xy | .
4.6. , -
, {x : x Z 1 6 x 6 12}.() R = {(x; y) : xy = 9};() S = {(x; y) : 2x = 3y};() R ;
() S .
4.7. .
.
() x , y ;
() x = 2y N ;
() x < y R ;
() x y .
4.8. ,
{(a; a); (b; b); (; ); (a; ); (a; d); (b; d); (; a); (d; a)
};
{a; b; ; d}. ?
4.9.
A ,
A:
() A | , R
: xR y, x -
y;
() A = Z, R : xR y ,
x y | ;() A| , xR y, x ,
y;
-
84 4.
() A = R2
, R : (a; b)R (; d) -
, a
2
+ b
2
=
2
+ d
2
.
4.10. R Z : xR y
, x
2y2 3. , R
.
4.11. -
:
() {1; 2; 3; 5; 6; 10; 15; 30} x - y;
() {1; 2; 3} X | Y .
4.12. R A =
= {a; b; ; d; e; f; g; h} . 4.6. - R
A.
a b c
d
g
e
h
f
4.6.
4.13. () -
: X Y
, ,
. X -
( ), Y ,
X Y ; X
Y , , X -
Y , , Y X.
-
85
: ,
, , . ,
.
A B
R A B. A = B, , R | - A.
( ), -
, .
R A
, xRx x A;, xR y y Rx x; y A;, (xR y y Rx x = y)
x; y A;, (xR y y R z) xR z x; y z A.
R
R -
P ,
1) R
P ;
2) R R;3) R
| ,
R P .
, R -
A .
x A E
x
= {z A : z Rx}: A -
A
1
; A
2
; : : : ; A
n
A, :
A = A
1
A2
An
A
i
Aj
= i 6= j: A
i
-
. R | A, -
A.
-
86 4.
, R
A . ,
, ,
.
| -
, .
R | A xR y,
x 6= y, x y. , - x y z, xR z
z R y, , x | -
y. : x y.
, -
. ,
x y, x - y .
.
, , . -
,
,
().
4.2. T1 =
.
4000123 1.2.83 2 ,
5001476 4.5.84 4 ,
5112391 21.3.84 17 ,
5072411 12.12.84 21 ,
5532289 15.8.83 4 ,
5083001 9.7.83 18 ,
5196236 21.3.84 133 ,
4936201 7.10.77 11 ,
, , .
, . 4.2 : -
, , , , -
. . 4.3
.
-
. 87
, -
. , . 4.2
, . 4.2 -
, . 4.3.
4.3. T2 =
.
. .
. .
n , A
1
;
A
2
; : : : ; A
n
-
A
1
A2
An
. n ,
A
i
, n-
.
, . 4.3 T2
A
1
A2
A3
A4
A5
, A
1
| ,
A
2
= A
3
= A
4
= A
5
= {, , , }. | (, , , ,
), ,
.
, -
, -
, : , .
, -
,
, .
. , (1, {, }) - . 4.4.
1.
(2, {, ., . .}).. . 4.5
-
88 4.
4.4. T3 = (1, {, })
2 ,
4 ,
17 ,
21 ,
4 ,
18 ,
133 ,
11 ,
4.5
.
. .
, -
, -
. , R S | ,
, R | -
A
1
Am
B1
Bn
, S |
A
1
Am
C1
Cp
. -
A
1
; A
2
; : : : ; A
m
. R S |
A
1
Am
B1
Bn
C1
Cp
, -
(a
1
; a
2
; : : : ; a
m
; b
1
; b
2
; : : : ; b
m
;
1
;
2
; : : : ;
p
),
(a
1
; : : : ; a
m
; b
1
; : : : ; b
m
) R, (a
1
; : : : ; a
m
;
1
; : : : ;
p
) |
S.
, (3, 2) . 4.6.
4.6
. . .
. .
2 ,
18 ,
4 ,
11 ,
133 ,
-
. 89
,
. , (1, = -
=) . 4.7.
4.7
.
5001476 4.5.84 4 ,
5083001 9.7.83 18 ,
2. (2, . .=).
. (. 4.8) -
2, , .
.
4.8
. . .
. .
, -
.
3. , -
:
R1 = (T2, {, ., . });R2 = (R1, . = . = );
. -, 2, -
, . . , . -
R1. ,
, ,
. R2 (. 4.9).
4.9
.
.
-
90 4.
4. :
R1 = (T1, =);
R2 = (T2,{, . .});R3= (R1, R2).
. 1 , -
, R1.
2 , , R2.
R1 R2 .
R1 R2, (. 4.10).
4.10
.
. .
4000123 1.2.83 2 ,
5196236 21.3.84 133 ,
4936201 7.10.77 11 ,
5. (, -
) -
, :
.
. -
.
R1=(1, =);
R2=(2, . .= . .=);
R3=(R2,.= .=);
R4= (R1, R3);
R=(R4,{, }).
-
5
, -
,
- .
| , -
.
,
. ,
, -
, 4.
.
-
. -
, . -
, -
, . ,
, -
.
, -
. -
, -
( ). ,
.
5.1.
R | A B. -
R
1 B A :
R
1= {(b; a) : (a; b) R}:
, ... ...
... ....
-
, .
-
92 5.
. R |
A B, S |
B C. R S
A C,
S R :S R = {(a; ) : a A; C aR b; b S b B}:
A C, B .
5.1. R | a | b, S -
b | .
: S R S S.. a | b, b | , a, ,
, . . a . ,
S R , a | . , S S |
a | .
5.2. , R S -
, . 5.1. , -
S R.a
b
R S
x
y
1 1
2 2
3 3
5.1. R S
. , , -
.
R = {(a; 1); (a; 2); (a; 3); (b; 2)} S = {(1; y); (2; x); (3; x)}: .
aR 1 1S y (a; y) S R;aR 2 2S x (a; x) S R;
-
5.1. 93
aR 3 3S x (a; x) S R;bR 2 2S x (b; x) S R:
. 5.2 .
a
b
x
ySR
5.2. S R
-
, . ,
.
.
:
A = {a1
; a
2
; : : : ; a
n
}; B = {b1
; b
2
; : : : ; b
m
} C = {1
;
2
; : : : ;
p
}:, R | A B, S |
B C. , M R
:
M(i; j) = (a
i
; b
j
) R;M(i; j) = (a
i
; b
j
) 6 R:, N S :
N(i; j) = (b
i
;
j
) S;N(i; j) = (b
i
;
j
) 6 S: b
k
B, ai
Rb
k
b
k
S
j
,
i- M k- . , j-
N k- .
, -
a
i
(S R) j
, P (i; j) P -
S R . i- M , j-
N , P (i; j) = .
-
94 5.
, P S R - :
P (i; j) = [M(i; 1) N(1; j)
[M(i; 2) N(2; j)
::::::::::::::::::::::::::::::::::::::::
[M(i; n) N(n; j):
P =MN .
5.3. R S | 5.2.
S R - R S.
. R A = {a; b} B = {1; 2; 3}
M =
[
];
A B
, .
, S | B = {1; 2; 3} C = {x; y},
N =
:
, P S R
P =
[
]
:
M , N | . -
P .
P (1; 1) M -
N . ,
P (1; 1) =
[
]
= ( ) ( ) ( )
= = :
-
5.1. 95
, M
, N .
: P (1; 1) = .
M -
N , ,
. , P (1; 2) = .
, M
N , P (2; 1) = .
, P (2; 2) = , M -
N .
,
N =
[
]:
5.4. R A = {1; 2; 3; 4; 5} -
:
R R , - R .
. R R
=
:
R R (x; z), xR y y R z - y A. R - R R R. , , , R R - , R. R
.
-
96 5.
5.2.
, A B. |
, -
.
A B -
, A
B. , -
a A (a; b). ,
, -
A, .
, . 5.3 , -
{a; b; } {1; 2}, (a; 1), (b; 1) (; 2).
a
b
c
1
2
5.3.
5.5. ,
A = {a; b; } B = {1; 2; 3} A B.
() f = {(a; 1); (a; 2); (b; 3); (; 2)};() g = {(a; 1); (b; 2); (; 1)};() h = {(a; 1); (; 2)}.
.
() f | , a -
B: 1 2.
() g .
() , -
b .
-
5.2. 97
5.6. :
() x | y ;
() Z, : {(x; x2) : x Z};() R, : {(x; y) : x = y2}
?
.
() ,
, .
() ,
x x
2
.
() | , , ,
: (2;
2) (2; 2) | .
, (x; y) x.
f | A B.
x A - y B, , (x; y) f , : y = f(x), -, f A B,
f(x) x f f ,
x.
, f : A B, , f A B. -
A , B |
1
f .
,
A f ,
. , f
B,
x A. f(A) :
f(A) = {f(x) : x A}: . 5.4 -
, A B.
1
, f A
B. | . .
-
98 5.
f
A B
f(A)
5.4. f : A B
f : A B, A B | - , .
-
, , .
5.5. y = f(x)
, f : R R, f(x) = x
2
, . 5.5. x
R. -
y (
R). , . . ,
(x; y) R R, y = f(x). ,, f(2) = 4, (2; 4) ,
.
5.7. g : R R, g(x) = 2 x. x = 2 x = 3. .
-
5.2. 99
. . 5.6. -
: g(2) = 0 g(3) = 1.y
x
(2,0)
(3, 1)-
5.6. g(x) = 2 x
.
f : A B | . - , ,
f(a
1
) = f(a
2
) a1
= a
2
a
1
; a
2
A. ,
a
1
6= a2
f(a1
) 6= f(a2
);
. . .
, .
f ,
,
. , b B a A, b = f(a). , - - f .
f ,
, .
5.8. , ,
. 5.7, , . -
.
-
100 5.
a
b
c
a
b
c
a
b
c
a
b
1
2
3
1
2
3
1
2
1
2
3
()
()
()
()
5.7.
.
() , 1 -
a, b. ,
, 2 .
() , -
. ,
.
() 1 a, b. -
, . -
,
.
() , .
() .
5.9. , h : Z Z, - h(x) = x
2
, .
. , -
f : R R, f(x) = x2, . -
, h
-
5.2. 101
. h, . 5.8,
.
-3-2 -1
1 2 3
1
4
9
y
x
5.8. h(x) = x
2
, Z.
, h ,
a
1
6= a2
, h(a
1
) = h(a
2
).
. , a
1
= 2 a
2
= 2. , -
, , -
h. -
. , -
1, .
5.10. , k : R R, - k(x) = 4x+ 3, .
. , k(a
1
) = k(a
2
), . .
4a
1
+ 3 = 4a
2
+ 3:
, 4a
1
= 4a
2
, a
1
= a
2
. , k |
.
b R. , a R, h(a) = b. , a a = 1
4
(b3)., k | .
k ,
| .
-
102 5.
5.3.
, f : A B | -. f
1.
, -
f
1: B A
.
f (a; b), b = f(a). f -
, f
1 (b; a), a = f
1(b).
, :
f(a) = b, f
1(b) = a. , -
.
5.11. 5.8 ?
.
, . , -
() .
.
k : R R, k = 4x + 3 (. 5.10). k :
-x
4
-4x
3
-4x+ 3
: 4 3 |
: 4 3, .
, :
1
4
(x 3)
4
(x 3)
3
x
, k
1: R R k1 = 1
4
(x 3). , -
. y = k(x), x = k
1(y).
, y = 4x+ 3. y, -
, x =
1
4
(y3). , k1(y) = 14
(y3) , - x , k
1(x) =
1
4
(x3), .
-
5.3. 103
, ,
, . : -
.
f : A B.. f ,
.
. .
, .
f : A B | . , f :
f =
{(a; b) : a A f(a) = b}:
:
f
1=
{(b; a) : a A f(a) = b}:
f , b B a A, f(a) = b. , - f a b .
, f
1 ,
B
A. , , , f
1
, .
, -
. , f
1| .
b B a A, - (b; a) f1. , (a; b) f , . . b = f(a). f .
f
. , f(a
1
) = f(a
2
). : (f(a
1
); a
1
)
(f(a
2
); a
2
) | f
1. f
1 ,
: a
1
= a
2
, f .
, f , . -
: ,
, .
5.12.
A = {x : x R x 6= 1} f : A A :
f(x) =
x
x 1 :
-
104 5.
, f .
. , f(a
1
) = f(a
2
).
a
1
a
1
1 =a
2
a
2
1 :
,
a
1
a
2
a1
= a
1
a
2
a2
;
a
1
= a
2
. , f .
b A | f . a A, : f(a) = b, . .
a
a 1 = b:
a,
a =
b
b 1 :
a =
b
b1 A, f(a) = b. f .
, , f , -
. , .
: f
1(b) = a ,
f(a) = b. , -
f ,
a =
b
b 1 :
, f
1: A A,
f
1(x) =
x
x 1 ;
. . f .
.
,
, .
f : A B g : B C | , g f A C (a; ), b B (a; b) f (b; ) g. b = f(a) a, f | . -
, = g(b) b (g
-
5.4. 105
). , = g(f(a)) -
a , ,
f g | .
, g f : A C , - (g f)(x) = g(f(x)). 5.13. : f : R R, f(x) = x2 g : R R, g(x) = 4x+ 3. g f , f g, f f g g.. R -
R.
(g f)(x) = g(f(x)) = g(x2) = 4x2 + 3;(f g)(x) = f(g(x)) = f(4x+ 3) = (4x+ 3)2 = 16x2 + 24x+ 9;(f f)(x) = f(f(x)) = f(x2) = x4;(g g)(x) = g(g(x)) = g(4x+ 3) = 4(4x + 3) + 3 = 16x+ 15.
-
.
. -
, -
sinx, log x, |x| . . , .
,
, -
. |
, , -
. , -
.
.
5.4.
f : A B | , A, B | - . , A n :
a
1
; a
2
; : : : ; a
n
. , |A| > |B|, f
1
.
, a
i
6= aj
, f(a
i
) = f(a
j
).
1
,
: 10 9 ,
. | . .
-
106 5.
, ,
i 6= j : f(ai
) 6= f(aj
).
B n :
f(a
1
); f(a
2
); : : : ; f(a
n
). , |B| > n, - : n = |A| > |B|. , a
i
; a
j
A, f(ai
) = f(a
j
).
5.14. 15 . ,
.
. A, -
12 B.
f : A B, . |A| = 15, |B| = 12, |A| > |B|. - f , . .
.
5.14 -
. 15 12 . -
, .
, -
, . ,
,
, , , -
. | -
. ,
, , f -
( A)
( B). -
.
5.15. -
,
,
?
. A | , B |
, -
, 33 . f : A B, -
: . ,
f() = (; ). B 33 33 = 1 089 . , |A| > |B| = 1089,
-
5.4. 107
, -
.
1 090
2
.
5.16. , 1, 2, 3, 4, 5, 6,
7 8 , ,
9.
. , 9.
{1; 8}; {2; 7}; {3; 6}; {4; 5}: A ( -
), B :
B =
{{1; 8}; {2; 7}; {3; 6}; {4; 5}}: f : A B, B, . -
, f(3) = {3; 6}. A . ,
9.
. -
f : A B, A B | . |A| > k|B| k,
f , k+1 .
,
f k , A
k|B| . 5.17. -
,
,
?
. f : A B | 5.15. , B 1 089 .
-
, , |A| > 4|B| = 4356. , - 4 357 .
2
, ,
. . | . .
-
108 5.
5.18. ,
, , ,
.
. x| , A| -
B = {0; 1}. f : A B :
f(a) =
{0; a x;
1; a x:
5 = |A| > 2|B|, , x, .
, a, b x.
,
. , - , a b
. x. , : a, b x|
. ,
, x .
, -
,
, . , -
A B. ,
, -
,
, -
.
5
5.1. R |
{1; 2; 3} {1; 2; 3; 4};
:
R =
{(1; 1); (2; 3); (2; 4); (3; 1); (3; 4)
}:
, S |
{1; 2; 3; 4} {1; 2};
-
5 109
:
S =
{(1; 1); (1; 2); (2; 1); (3; 1); (4; 2)
}:
R
1, S
1 S R. ,
(S R)1 = R1 S1:
5.2. R | ... ..., S |
... ... . -
: R
1, S
1, RS, S1 R1 RR.
5.3. , R |
A, R
1 -
A.
-
R R
1?
5.4. R S M N ,
M =
[
] N =
:
MN . -
?
5.5. A = {0; 2; 4; 6} B = {1; 3; 5; 7}. - A B
, A B?
()
{(6; 3); (2; 1); (0; 3); (4; 5)
};
()
{(2; 3); (4; 7); (0; 1); (6; 5)
};
()
{(2; 1); (4; 5); (6; 3)
};
()
{(6; 1); (0; 3); (4; 1); (0; 7); (2; 5)
}.
, -
?
5.6. , -
Z, ,
, .
() f(n) = 2n+ 1;
-
110 5.
() g(n) =
{n
2
; n ;
2n; n ;
() h(n) =
{n+ 1; n ;
n 1; n :5.7. :
() f : Z Z, f(x) = x2 + 1;() g : N N, g(x) = 2x;() h : R R, h(x) = 5x 1;
() j : R R, j(x) ={
2x 3 x > 1;x+ 1 x < 1;
() k : R R, k(x) = x+ |x|;() l : R R, l(x) = 2x |x|. , -
, (|x| x, x x > 0 x x < 0).
5.8. , , -
x , -
x, : x.() A = {1; 0; 1; 2} f : A Z -
: f(x) =
x
2
+1
3
:
f .
() , g : Z Z,
g(n) =
n
2
, .
5.9. f : A B : f(x) = 1 + 2x
, A
, 0,
B | 1. ,
f .
5.10. f : R R g : R R :
f(x) = x
2
g(x) =
{2x+ 1; x > 0;
x; x < 0:
: f g, g f g g.
-
5 111
5.11. f : A B g : B C | . , () f g , g f ;() f g , g f ;() f g , (g f)1 = f1 g1.
5.12. () ,
- ?
() ,
:
?
() -
52 ,
?
() -
52 ,
?
5.13. , 79 ,
2 .
() , ,
, . .,
,
| .
() ,
.
5.14. S = {1; 2; : : : ; 20}.()
S,
22?
() , 11 S,
-
- .
(: f , -
-
. , f(12) = 3.)
-
112 5.
R A B
R
1; -
B A : R
1=
{(b; a) : (a; b) R}.
R | A B, S | -
B C. -
R S A C,
:
S R = {(a; ) : a A; C aR b; b S b B}: M N | R S -
. MN
S R.
, A B, -
f A B,
A B.
f : A B A - B. A f ,
B | f . y = f(x),
, y B | f , x. y x -
f .
f B:
f(A) =
{f(x) : x A} ( ).
f : A B ( ), f(a
1
) = f(a
2
) a1
= a
2
a
1
; a
2
A.
f : A B , - . ,
b B a A, f(a) = b.
, , ,
.
f , -
f . f : A B
-
. 113
, . f
f
1: B A. f(a) = b, f1(b) = a.
, f : A B | -, A B,
|A| > |B|, f . |A| > k|B| k, f k+1 .
.
,
.
, -
, -
.
-
, -
. , -
, .
, -
,
.
C =
{; ; ; : : : ;
}| -
, P
{0; 1; 2; : : :
}. S (-
) . , | -
S, .
, -
:
har : S C, har(s) | s.
rest : S S, rest(s) | , s .
addhar : CS S, addhar(; s) | , - s .
len : S P, len(s) | s.
-
114 5.
,
, . -
, ,
| -
.
1.
har(s),
len(rest(s))
addhar
(
har(s); addhar
(; rest(s)
)),
s = .
.
har(s) = har() = ;
len(rest(s)) = len(rest()) = len() = 2;
addhar
(
har(s); addhar
(; rest(s)
))=
= addhar
(; addhar
(; rest()
))=
= addhar
(; addhar(; )
)=
= addhar(; ) = .
2.
addhar(har(s); addhar(;rest(s)));
s | , | .
. ,
s.
3. third : S C - .
third har rest.
. s -
, s
.
third(s) = har(rest(rest(s))):
-
. 115
4. ,
reverse2 : S S, 2 .
. s | . -
har(rest(s)), | har(s).
rest(rest(s)). -
, reverse2(s)
:
addhar(har(rest(s)); addhar(har(s); rest(rest(s)))):
5. , -
s = .
Input s
begin
u
:
= ;
t
:
= s;
i
:
= 0;
while i < len(s) do
:
=har(t);
t
:
=rest(t);
u
:
=addhar(; u);
i
:
= i+ 1;
end
Output u
?
. , t, u
i while
. 5.1
5.1
t u i i < 4?
0 | 0
1 1
2 2
3 3
4 3
.
-
116 5.
, -
. , rest
s = , reverse2 -
2. , -
,
. -
. ,
.
, rest -
:
rest : S S, : s S s 6= ,
rest(s) | , s, .
, -
. , ,
rest(rest(s)) 1 ,
rest rest | s S len(s) > 1.
-
6
, -
. ,
,
. , -
( 3) ( 5).
, -
:
.
.
, :
,
.
. -
-
, .
,
:
(x
1
+ x
2
+ + xk
)
n
:
, -
.
,
.
, -
.
6.1.
.
1.
: ,
.
.
?
-
118 6.
2. ,
. -
6 9 .
?
3. 3 4?
. -
, .
4+2+3 = 9 ,
.
6 ,
,
. ,
, , 69 = 54. -
.
, A B | -
, n
1
A, n
2
-
B,
A B n
1
+ n
2
.
, -
k n
1
, n
2
| -
, . ., n
k
,
k
n
1
n2
nk
.
, , | -
. , A B
, |A| = n1
, |B| = n2
; A
B , , -
. ,
, |A B| = |A| + |B|, . . A B n
1
+n
2
. , n
1
+n
2
A B.
-
. A
1
n
1
, A
2
| n
2
, . .
k
A
1
A2
Ak
,
|A1
| |A2
| |Ak
|. .
, -
. , , -
-
6.1. 119
.
, 3, -
| 4. , -
(
3), 10 10 .
,
1 10 10 = 100. . 100. ,
, 100 + 100 = 200 -
, 3 4.
6.1. .
, .
?
.
, 3 4 = 12 -
1
. 3 2 = 6 ., 4 2 = 8 -. ,
, ,
12 + 6 + 8 = 26.
6.2.
( ). , -
. -
?
. 33
. -
33 33 33 = 35 937. , 10 10 10 = 1 000. ,
, : 35 937 000
.
1
: -
.
, . | .
.
-
120 6.
6.2.
, -
: (A), - (B)
(C).
?
. , ,
. ,
, . . AA? ,
? ,
AB BA ?
-
.
1. .
9 : AA, AB, AC, BA, BB, BC,
CA, CB CC.
2. ,
. | 6 : AB, AC, BA, BC,
CA CB.
3. , .
| 6 : AA, AB, AC, BB, BC
CC.
4. , , ,
,
: AB, AC BC.
, -
.
, .
. , -
x
1
; x
2
; : : : ; x
k
X k.
k n -
, , (n; k)-. -
, .
,
, . -
, -
.
, .
-
6.2. 121
(n; k)- - (n; k)-, ;
(n; k)- - (n; k)-, -
;
(n; k)- - (n; k)- ;
(n; k)- (n; k)- .
(n; k)--
. -
n .
,
, . . k
, , , -
(n; k)- n
k
.
6.3.
N . (+ ), N 1 . - ?
. | 0 1. -
N . , , -
, ,
, , .
(2; N)- . -
, 2
N
.
, :
000000 00 + 000000 00; 0. ,
(2
N 1) . (n; k)-
1
: P (n; k). . -
n . -
,
1
A
k
n
. | .
.
-
122 6.
(n 1) . - | (n 2) , k- , (nk+1) . .
P (n; k) = n(n 1)(n 2) (n k + 1):
, -
1 n n -
n!.
P (n; k), , -
.
P (n; k) = n(n 1)(n 2) (n k + 1) =
= n(n 1)(n 2) (n k + 1)(n k)(n k 1) 2 1(n k)(n k 1) 2 1 =
=
n(n 1)(n 2) (n k + 1)(n k)(n k 1) 2 1(n k)(n k 1) 2 1 =
=
n!
(n k)! :
, (n; k)-
P (n; k) =
n!
(n k)! :
6.4.
, : , , , ,
,
.
. ,
,
. -
P (6; 4). ,
P (6; 4) =
6!
(6 4)! =6!
2!
=
6 5 4 3 2 12 1 :
:
P (6; 4) =
6 5 4 3 62 6162 61 = 6 5 4 3 = 360:
-
6.2. 123
, . . ,
.
(n; k)-
2
C(n; k).
.
: (n; k)-
P (n; k) =
n!
(nk)! . -
, P (n; k), , C(n; k).
, .
. n = 4, k = 3. -
A = {1; 2; 3; 4}, . (4; 3)- |
, ,
, . , -
{1; 2; 3} (4; 3)- . - {2; 1; 3}, ( ),
( ). -
? -
, .
(n = 4, k = 3) ,
. . -
.
(n; k)- , . . -
B A, |B| = k |A| = n. (n; k)- ?
, (k; k)-
! (, .)
3
:
P (k; k) =
k!
(k k)! = k!:
, (n; k)- -
k! (n; k)- . ,
C(n; k) =
P (n; k)
k!
=
n!
(n k)! k! :
2
C
k
n
, ,
, :
`n
k
. | . .
3
, 0! = 1. | . .
-
124 6.
6.5.
.
?
. (7; 3)- -
. :
C(7; 3) =
7!
(7 3)! 3! =7!
4! 3!
=
7 6 5 4 3 2 1(4 3 2 1)(3 2 1) =
=
7 6 5 64 63 62 61(64 63 62 61)(3 2 1) =
7 66 563 62 61 = 35:
, 35 .
, , . -
, , , -
.
, ,
, - -
.
, , ,
, a, .
, , , -
||, : ||. , - , , |
, , , -
, . , ,
( ),
,
. , -
,
. ,
, (7; 2)- , . . -
C(7; 2). ,
, | , -
. 7 6 . , , -
. , 7 6 2. ,
-
6.2. 125
7 62
=
7 6 5 4 3 2 1(2 1)(5 4 3 2 1) =
=
7!
5! 2! =7!
(7 2)! 2! = C(7; 2):
(n; k)-
(k n ), , n 1 - k . , (n 1) + k . , (n; k)- -
(n 1) (n+ k 1). , (n; k)-
C(n+ k 1; n 1) = (n+ k 1)!(n+ k 1 (n 1))! (n 1)! =
(n+ k 1)!k! (n 1)! :
6.6. , -
?
. -
, . . .
, -
( 6
) , . . (6; 5)- -
. ,
C(6 + 5 1; 6 1) = C(10; 5) = 10!5! 5!
= 252:
. 6.1 -
k n- ,
.
6.1
n
k
(n+ k 1)!
k! (n 1)!
n!
(n k)!
n!
(n k)! k!
, -
, -
.
-
126 6.
4
-
49 -
. , ,
, -
.
. -
|
49 . (49; 6)-
49!
(49 6)! 6! =49!
43! 6!
= 13 983 816;
: 1 13 983 816. -
, , .
, , , -
, .
, , ,
, ,
.
, ,
. ,
, $1 ( ), ,
( ) , -
$10. $10.
, -
: -
. C(6; 3) -
,
, C(43; 3) -
. , $10,
C(6; 3) C(43; 3) = 6!3! 3!
43!40! 3!
= 246 820:
|
, . .
246 820
13 983 816
157
0; 018:
4
. | . .
-
6.2. 127
6.7. , , -
.
12 ?
() , ;
() , ;
() , , ?
.
C(12; 5) =
12!
7! 5!
= 792
.
() , -
-
.
C(10; 3) =
10!
7! 3!
= 120
. , 120
.
() ,
.
C(10; 5) =
10!
5! 5!
= 252
, ,
.
()
, , .
C(10; 4). ,
. , 2C(10; 4) , , .
, -
792
: (), () (). ,
, ,
792 120 252 = 420:
-
128 6.
6.3.
C(n; k)
(a+ b)
n
. ,
(a+ b)
3
= (a+ b)(a+ b)(a+ b) =
= aaa+ aab+ aba+ abb+ baa+ bab+ bba+ bbb =
= a
3
+ 3a
2
b+ 3ab
2
+ b
3
:
, -
, , -
. , ,
a b.
, C(3; 2) = 3
, b ( a).
-
: C(3; 0) = 1, C(3; 1) = 3, C(3; 2) = 3 C(3; 3) = 1.
C(n; k),
, , 0! = 1.
, -
.
, (a + b)
n
,
a
nkb
k
( k
0 n) b, k , a,
(n k) . C(n; k) k n, C(n; k)
a
nkb
k
k = 0; 1; : : : ; n. ,
(a+ b)
n
= C(n; 0)a
n
+C(n; 1)a
n1b+C(n; 2)a
n2b
2
+ +C(n; n)bn: . -
C(n; k) .
-
(. . 6.1).
C(0; 0)
C(1; 0) C(1; 1)
C(2; 0) C(2; 1) C(2; 2)
C(3; 0) C(3; 1) C(3; 2) C(3; 3)
C(4; 0) C(4; 1) C(4; 2) C(4; 3) C(4; 4)
C(5; 0) C(5; 1) C(5; 2) C(5; 3) C(5; 4) C(5; 5)
: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
C(n; 0) C(n; 1) : : : : : : : : : : : : : : : C(n; n 1) C(n; n)
6.1.
-
6.3. 129
(n + 1)- -
,
(a+ b)
n
.
-
,
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
::: ::: ::: ::: ::: :::
