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Маака Iа а ај
Рак Вкћ, .
2010-11. .. .
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Садржај
I .............................................................................................................................. 5 1. .......................................................................................................... 5
1.1 .............................................................................................................. 6 1.2 ............................................................................................................. 6
1.3 .............................................................................................................. 8 1.4 ...................................................................................... 10 1.5 .......................................................................................................... 11
2. .......................................................................................................................... 13 2.1 .......................................................................................... 13 2.2 ............................................................................................................ 16 2.3 ................................................................................................................. 17
3. .................................................................................................................... 18 3.1 ............................................................................................................ 18
4. .................................................................................................................. 21
4.1 ............................................................................................................ 21 4.2 ........................................................................................... 24 4.3 ..................................................................................... 25
4.4 ................................................................................................................. 26 II .......................................................................................................................... 28
5. ......................................................................................................... 28
5.1 ................................................................................... 28 5.2 .......................................................................................................... 30 5.3 ..................................................................................................... 34
6. .................................................................................................................. 36
6.1. ....................................................................................................... 36
6.2. ....................................................................................................... 38 6.3. ....................................................................................... 40 6.4. ................................................................................................. 44
7. ....................................................................................................................... 46 7.1. ...................................................................................... 46 7.2. ................................................................................... 48 7.3. ......................................................................................... 49
8. ....................................................................................................................... 51 III .................................................................................................................. 55
λ. ................................................................................................................... 56
λ.1. , ............................................................................. 57
λ.2. .......................................................................................... 59 λ.3. .................................................................................... 60
λ.4. ........................................................................................ 61 λ.5. .................................................................................. 62
10. .................................................................................................................... 62
10.1. .................................................................... 63 10.2. ....................................................................... 64
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10.3. ...................................................................................... 66 10.4. .............................................................................................................. 67
11. .......................................................................................................... 67 11.1. ................................................................................. 68
11.2. .......................................................................... 70
11.3. ....................................................................................... 73
11.4. .................................................................................. 75 11.5. ...................................................................................... 77
IV ................................................................................................................... 80 12. ......................................................................................................... 80
12.1. .................................................................................... 81 12.2. ........................................................................ 83
13. ............................................................................................................. 86 13.1. ............................................................................................... 87 13.2. ..................................................................................... 89 13.3. ........................................................................................................... 92
13.4. .................................................................................................... 93
14. ................................................................................................................ 93
14.1. ........................................................................................... 94 14.2. ...................................................................................................... 97 14.3. ................................................................................................ 98
15. ............................................................................................ 102 15.1. ....................................................................................................... 102 15.2. ............................................................................................................ 105
V ....................................................................................................................... 107 16. Ј ............................................................................................................ 108
16.1. ..................................................................................... 109 16.2. ............................................................................................... 113 16.3. ................................................................................. 118
17. ....................................................................................................... 121 17.1 ................................................................................................... 121 17.2. .................................................................................. 123 17.3. ............................................................................. 126
VI ..................................................................................................................... 130 18. ................................................................................................. 130
18.1. ........................................................................................ 130 18.2. ................................................................................. 133 18.3. Ј ............................................................................................ 137
1λ. ............................................................................................................... 139
1λ.1. ....................................................................................... 139
1λ.2. ........................................................................................ 145 1λ.3. Ј .......................................................................... 148
VII ................................................................................................................ 153 20. .......................................................................... 154
20.1. Ј ..................................................................................... 154 20.2. ..................................................................... 156
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20.3. .................................................................................................... 159 21. .................................................................................... 162
21.1. ................................................................................. 162 21.2. ....................................................................................... 163
21.3. ........................................................................................ 164
VIII ..................................................................................................... 170
22. .............................................................................................. 170 22.1. ................................................ 170 22.2. ................................................................... 173 22.3. ...................................................................................................... 175
23. .......................................................................................... 178 23.1. .......................................................................................... 179 23.2. ....................................................................................... 181
24. ........................................................................................... 184 24.1. ............................................................................... 184 24.2. ................................................................................................. 188
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I
1. А , . , , , .
μ , , . , .
1. (. )ν
2. (. ).
, . , . , „“ , . ,
, , . , , .ђ, .
, , , . , . ђ
, . ђ , , ђ .
, . , , . „ “ , , . .
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() = (. „“), () = („-“). 1 0. .
1.1
1.1.1 „2+3ο5“ . „2+3ο6“ .
1.1.2 „ 180њ“ .
1.1.3 „ “ .
1.1.4 „ “ , , . , 4 6 , .
1.1.5 „Ј “ . , , . , , . , , !
1.2
, („“) („“). , , , , ,
.
( „-“) . , .
A, B A B („ “, „ “) . , . , „ “ „ “ „ “.
A, B A B („ “, „ “) . , . , „ “ „ “ „ “.
, :
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, μ ,
.
A, B A B („ B„) . . , „ “ .
A, B A B (. „ B“) , B . , .
, „ “ „ “ „ “.
A, B A B (. „ B“) , . (),
().
, μ
, , . . .
, , .
1.2.1. :
) (a b) a b; ) (a b) a b.
: )
a b a b (a b) a b a b T T T TT T T T
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T T T T T T T T T
1.3
1.3.1 μ
() (2,3456 – 1,2345 = 10/9); (b)
5
3
7
5
13
11
19
17;
(c)
12
1
4
1
3
1
6
1
3
1
2
1; (d)
11
9
15
13
15
13
19
17.
1.3.2 μ
(a)
2
22
n
m
n
mnm ; (b)
1
1
2
2
1
1
n
m
n
m
n
m
n
m.
1.3.3 (a) ; (b) ; (c) ; , .
, μ
(a) A & B = (A B) (A B);(b) A B = B A;(c) A B = (A B) (A B).
1.3.4 :
(i) A (B A);
(ii) ((A B) A) A;(iii) (A B) ((B C) (A C)).
(i), .
, (. B A). , (. = T) ( ). ,
, B (B = T) ( = ). , , , . . .
1.3.5 :(iv) A B A;(v) A B B;(vi) (A B) ((A C) (A B C)).
(vi) μ
B C A B A C A B C (A C) (A B C)
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, , (vi) .
1.3.6 :
(vii) A A B;(viii) B A B;(ix) (A C) ((B C) (A B C)).
1.3.7 :
(x) (A B) (A B);(xi) (A B) (B A);(xii) (A B) ((B A) (A B)).
1.3.8 :
(xiii) A (A B);(xiv) (A B) ((A B) A);(xv) (A B) ((A B) B).
1.3.9 , . .
, μ
abba a, b „“;
cbacba )()( a, b, c ;
)()()( cabacba a, b, c
.
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џ 1854. „ “
, , ,, , / .
.
1.4
.
, , , . , .
:1.
ν
2. A B (Modus ponens) Bν
3. ()
. , , |- A,
„ “.
ђ . . , . .
:
1. (Modus ponens). B ,
B;2.
(Demonstratio per enumerationem – ). C B C , (A B) C;
3.
(Deductio ad absurdum – ђ ). B B , A.
. , (1) B
. , B = ,
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. , B = . , .
:
1.
A |- B A ( B );2. A B |- (B C) (A C) ( AB ... );3.
A B, B C |- B (A C) ( AB BC ... );4.
|- (A B) (B A) (- )ν 5. |- (A (A B)) (A B) (- );6. |- ((A B) A) ((A B) B);7. |- (((B C) C) (A C)) (B (A C));8. |- (A (B C)) (B (A C));9.
|- (A B) (A) (B) ( );10.
|- (A B) (A) (B).
. (1) (i), . (2) (iii), . (3) (ii), . ђ, , , ђ ђ , .
, () (). , „ “ μ
(xN)(yN)(x < y), , x = 1.
1.4.1. ς
) (xR)(yR)(x < y); ) (xR)(yR)(x < y);) (aZ)(bZ)(a + b = 0); ) (aZ)(bZ)(a + b = 0); R , Z .
1.5
Ј . , , . , ,
. , .
.
A B ()μ
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.
A B, μ
A B μ
A B μ
1.5.1 : A (B C), A (B C).
1.5.2 , :
(1) A (B C) = (A B) ( C); (2) A (B C) = (A B) (A C).
А B
А
B
B
A
B
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(Claude Elwood Shannon, 1916. – 2001), .
21- - . ђ 1939.
1λ48. ,
.
2. , . . , . .
џ (1845 – 1λ18), ђ .
1874. 10 . .
ђ, , . . .
, , .
2.1
S , . - , .
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, B -- , Њ. :
S = {, , ј}, = {, }, B = {, , }.
, |S| = 3, |E| = 2, |B|= 3. , ђ . „“ B,
B. ђ, „“ S B, „“ S E, „“ E, B. μ
S, B; S, E;
E, B.
. ,
P = S B
S B x P x S x B.
P = {x| x S x B},
. P S B. P = {}. P E , . P E =.
. ,
U = S B
S B x U x S x B.
U = {x| x S x B},
. U S B. U = {, , ј, , }.
() μ
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.
S U
S U S U. x S x U.
2.1.1 А B . A = B A B B A.
2.1.2 А B . :1.
A A B B A B;2. A B A A B B.
. ,
R = S \ B
S B x R x S x B.
R = {x| x S x B},
. R S B. R = {, ј}.
S U U
S. U \ S. U S' = {x| x S}, . S S.
. S,
(S) {A| A S}.
(S) = {, {}, {}, {ј}, {, }, {, ј}, {,ј}, S}.
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(S) 23 .
A, B ђ .,
A B = {(x, y)| x A y B}.
2.2
2.2.1 A (B C) = (A B) (A C) A, B, C .
2.2.2 30 ђ μ , . 1λ ђ, 14, 13 ђ . 6 , 5, 7. 2 . ς
P, L, I.
2 . 4, 3, 5 , , .
, 8, 5, 3. , 30.
2.2.3 μ
A, B, C (A B)’ ο A’ B’ (A B)’ ο A’ B’.
. , . , .
A B A B (A B)’ A’ B’ A’ B’
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2.2.4 ,
μ A (B C) = (A B) (A C), .
. ђ ( x, y). ( x, y) A (B C) x A y (B C) x A ( y B y C). ,
(x A y B) (x A y C) (( x, y) A B) (( x, y) A C) ( x, y) (A B) (A C). , , .
2.3
2.3.1 A, B, C μ A C = {a, b, c, d, e, f, g},B C = {d, e, f, g}, A C = {e}, B C = {e, f}, A \ B = {a, b, c}, B \ C = {d}.
2.3.2 M, N, P μ M N = {1, 2, 3, 4, 5, 6},N P = {2, 4, 5, 6, 7, 8}, P M = {1, 3, 5, 6, 7, 8}, M N = {5, 6}, N \ P = {7, 8}.
2.3.3 B μ
(a) A B A B = A; (b) A B A B = B;(c) A (A B) = A; (d) A (A B) = A.
2.3.4 A B S.
() A B = |A B| = |A| + |B|;
(b) |A B| = |A| + |B| - |A B|;(c) |A \ B| = |A| - |A B|.
2.3.5 S (a) 10; (b) 12; ,
(S)?
2.3.6 () ν (b) .
2.3.7 A, B, C
(a) A \ (B C) = (A \ B) (A \ C); (b) A \ (B C) = (A \ B) (A \ C).
2.3.8 , (a) A (B \ C) = (A B) \ (A C);(b) A \ (B C) = (A \ B) C;(c) ( \ B)’ ο (B \ A)’ν
(d) A (B C) = (A B) (A C);(e) A (B \ C) = (A B) \ (A C).
2.3.9 B
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A B = (A \ B) (B \ A).(a) A = {a, b, c} B = {b, c, d, e}, A B.(b) .
(c) A B = (A B) \ (A B).
2.3 (1) A = {a, b, c, d, e}, B = {d, e, f}, C = {e, f, g}; (2) M = {1, 3, 5, 6}, N= {2, 4, 5, 6}, P = {1, 3, 5, 6, 7, 8}. (5a) 2
10 = 1024.
3. ј
, . ђ ђ .
, B A B. , .
A B:
= {(b,1), (b,2), (c,2)},
. (b,1), (b,2), (c,2) , b1, b2, c3. , -, , B, - (
). . (b,1) (1,b).
.
3.1
3.1.1. . ђ .
: = {(a,a), (a,e), (b,b), (b,d), (b,e), (c,d), (d,c), (e,c)}.
3.1.2 = {a, b, c} B = {1, 2}. ς
: A B = {(a,1), (a,2), (b,1), (b,2), (c,1), (c,2)}. |A B| = |A||B| = 32 = 6 , ђ . 26 = 64, , 63
( ). B A 63 . , 126 .
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3.1.3 Z ο д…, -1, 0, 1, 2, 3, …ж. ђ (m,n) , m < n, .
= {(m,n) Z Z | m < n}.
Z.
3.1.4 , , xy „ “.
3.1.5 R. = {(x,y) | x, y R, x2 + y2 = 1, y > 0}. R. , , (0,0) 1 .
. .
( ) μ
μ (x) xx.
μ (x,y) xy yx.
μ (x,y) xy yx x = y. μ (x,y,z) xy yz xz.
, , , , .
, ђ. ђ, m n ђ , .
, , . „ , “.
, .
, , . () . . „ “ , „“ , „“ .
, , . , . ђ , . Ј ,
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, () , . .
[x] = {y | xy}, , . μ
1.
x A, [x] ;2. в [x], [x] = [y], x, y A;3.
x, y A, [x] = [y], [x] [y] =;4. .
.
.
3.1.6 N ο д1, 2, 3, …ж. , 3, 0, 1, 2.
(x,y) , .
xy x y (mod 3),
. , . , , , , . μ
Д0Ж ο д3, 6, λ, …ж – . Д1Ж ο д1, 4, 7, …ж – .
Д2Ж ο д2, 5, 8, …ж – .
[3] , [4] , . . {[0], [1], [2]}.
[0],. . , . ,
xy x – y 0 (mod 3),
. , .
3.1.7 . xy , . , . , , , . . [2] [1] д2, 4, 6, …ж, д1, 3, 5, …ж.
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4. ј . , , .
f: A B,
f(a) = 1, f(b) = 2, f(c) = 2.
f = {(a,1), (b,2), (c,3)ж , .
μ
(xA)(!yB) y = f(x),
. .
f X Y ђ X Y, X , .
D(f) = X, (x,y1), (x,y2) f y1 = y2.
, . , .
(Leibniz, 1646-1716)
„“ .
(Clairant) 1734. f(x), (Euler, 1707-1783) .
(Dirichlet, 1805-185λ), , .
4.1 , „“ Y, . , . , μ
(yY)(xX) y = f(x),
. .
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, 1-1 B,. , . , μ
f(x1) = f(x2) x1 = x2 .
4.1.1 , 3 B . , f . ђ
, 2 B , b c .
4.1.2
1
4
4
3
4
2
3
1f , . f(1) = 3, f(2) = 4, f(3) = 4, f(4) = 1
{1, 2, 3, 4}2 = {1, 2, 3, 4}{1, 2, 3, 4}. , .
4.1.3 Z Z f = {(n, 2n-3) | n Z}. f(n) = 2n - 3, n . , 2n – 3 n.
, , m ο 2 n. , 2n – 3 = m n = (m – 3)/2, .
ђ, ,
f(n1) = f(n2) 2n1 – 3 = 2n2 – 3 n1 = n2 ,
. .
4.1.4 f = x2
. , . f: R [0,+].
. , .
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f = x2 , . f: x x2 , .
, , , -. , B, , -. .
ο 2, = 4 (2, 4) -.
ђ, , (+2)2
= (-2)2
= 4, . 2 4.
, .
4.1.5
1
4
2
3
4
2
3
1g , .
. ђ μ
2
4
1
3
3
2
4
11g .
, .
4.1.6. .
4.1.7 f(x+1) = x - 2 .
!
ђ, , .
: ο 1 f(2) = -1, ο 2 f(3) = 0, ο 3 f(4) = 1, ο 4 f(5) = 2, ο 5 f(6) = 3,
... .
x+1 = t, x = t – 1, f(t) = t – 3,
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f(x) = x – 3. t. , .
, , . f(x) = x – 3, . , (4.1.7) ,
.
r f(x) = x – 3, x – 3 = r. x = r + 3, r
f -1(x) = x + 3.
f -1 , . + 3, . , .
4.2
f -1
f -
„“. , . .
, („“)
( B) ( ).
„“ ( B) , !
3.
, („1-1“ ),
(2 B) (b, c ). , B A, .
( 4.1.6), , . .
4.2.1 X Y f : X Y. S X, f(S) {f(x) | x S} S f.
A = {a, b, c} f(A) = {1, 2}.
4.2.2 f f(A B) = f(A) f(B), A B D.
: f . B D.
y f(AB). y = f(x) x AB. y f(A)f(B).
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f(AB) f(A) f(B). , y f(A) f(B). y f(A) y f(B). ,y = f(a) a A, y = f(b) b B. f(a) =f(b), a = b. , y f(AB). f(A) f(B) f(AB). f(AB) = f(A) f(B).
, f(AB) = f(A) f(B) D. . , x, y D f(x) = f(y) x y. ο {x} B = {y}. AB =, f(AB) =. ђ, f(A) f(B) = {f(x)} . f(AB) f(A) f(B), . , .
4.3
f : A B g : B C, .
h : A C,
h(x) = (g o f)(x) = g(f(x)).
, g.
μ
h(a) = (g o f)(a) = g(f(a)) = g(1) = ,h(b) = (g o f)(b) = g(f(b)) = g(2) = ,h(c) = (g o f)(c) = g(f(c)) = g(2) = .
4.3.1
14
43
32
21f ,
24
43
12
31g .
) f o g ν ) g o f ν ) f 2 = f o f ν ) g2.
) ) μ
f o g g o f
(f o g)(1) = f(g(1)) = f(3) = 4 (g o f)(1) = g(f(1)) = g(2) = 1
(f o g)(2) = f(g(2)) = f(1) = 2 (g o f)(2) = g(f(2)) = g(3) = 4
(f o g)(3) = f(g(3)) = f(4) = 1 (g o f)(3) = g(f(3)) = g(4) = 2
(f o g)(4) = f(g(4)) = f(2) = 3 (g o f)(4) = g(f(4)) = g(1) = 3
)
34
13
22
41gf ν )
34
23
42
11f g ; )
24
13
42
312f .
. , .
4.3.2 f(x) = 2x + 1, g(x) = 2x – 3, h(x) = 3x – 1,
(x R). ) ,f o (g o h) = (f o g) o h. ) f o g o h.
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: ) [f o (g o h)](x) = f((g o h)(x)) = f(g(h(x))) [(f o g) o h](x) = (f o g)(h(x)) = f(g(h(x))). ) f(g(h(x))) = f(g(3x – 1)) = f(2(3x – 1) – 3) =2(2(3x – 1) – 3) + 1 = 12x – 9.
4.4
4.4.1 f : R R , :
) f(x) = 2x2 – x + 1 ; b) f(x) = x3 + 2x ; c) f(x) = |x| – 3x + 2 ; d)3
2)(
x
xxf .
4.4.2 A = {1, 2, 3}.a) . b) .
4.4.3 , f : R R :
a) 23
12
)2
1
(
x
x
x
x
f ; b) 32
23
)2
1
(
x
x
x
x
f ; c) 23
32
)1
2
(
x
x
x
x
f ;
d) xx
f xf 3)1
()(2 ; e)xx
f xf 1
)1
(2)(3 ; f) 2)1
(3)( xx
f xf .
: ) tx
x
2
1
t
tx
1
21,
21
213
11
212
)(
t
tt
t
tf t
t
45
51
,
x
xxf
45
51)(
. d) x = t, t
tf tf 3)
1()(2 ;
x = 1/t, ttf tf 3
)()1
(2 .
tvu 32 t
uv 3
2 , )(tf u , )1(t
f v t
. t
tu 12 ,
xxxf
12)( .
4.4.4
1
4
2
3
3
2
4
1f ,
1
4
3
3
4
2
2
1g
1
4
4
3
3
2
3
1h .
a) f 2 , g 3 h 3 ; ς
b) e je u v, f o u = h v o g = h .c) (f o g)-1 = g -1 o f -1.
4.4.5 f : X Y A, B X. :a) f(AB) = f(A) f(B);b) f(AB) f(A) f(B);c) f(A\B) f(A) \ f(B), .
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4.4.6 f : X Y, S Y f -1(S) = {x X | f(x) S}. A, B Y.μ
a) f-1
(AB) = f -1(A) f -1(B);b) f -1(AB) = f -1(A) f -1(B);c) f -1(A\B) = f -1(A) \ f -1(B).
4.4.7 A, B ђ f : A B. A B, |A| = |B|. :a) ν
b) , . Z N;c) , . Q N;d) -1 1, . R (-1, 1).
: d) , x
xxf
1)( R (-1, 1).
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II , .
μ (R), (I), , . (Q).
(Z), (N).
.
5. Цј ј N ο д1, 2, 3, 4, … ж Z = {0, 1, -1, 2, -2, 3, -3, … ж,
f : N Z,
. , () .
N Z, kard N = kard Z, |N| = |Z| =0.
0 – , „“ , „“ .
5.1
ђ 5.1.1:
(Z#) , . S Z#, x S x y y S. , .
5.1.2: S Z#, no
S. S μ 1. n no , n Z
#, n S n + 1 S;2. no m < n , n Z
#, m S n S.
Tada {n Z# | n no } S.
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5.1.3
n μ2
)1(...321
nn
n .
.
, n = 1. 1, 1,
2
)11(1 , . ђ 1. ! , , n = 4.
2
)14(44321
,
10, . ! ђ, , .
.
(n no) S(n) S(n+1), μ
S(n) :2
)1(...321
nn
n
S(n+1) :2
)2)(1()1(...321
nnnn .
S(n+1) S(n)
(n+1). )1(2
)1(
n
nn, ђ
2
)2)(1( nn, . ,
, .
, .
5.1.4 (n 3) 2n + 1 2n .
μ
n = 3, 23 + 1 23 , , 7 8. n = 4, 24 + 1 24 , , λ 16. n = 5, 25 + 1 25 , , 11 32.
(n no = 3) S(n) S(n+1), μ
S(n) : 2n + 1 2n
, S(n + 1) : 2(n+1)+ 1 2n+1
. S(n+1) S(n). 2(n+1)+ 1 = 2n + 2 + 1 = (2n + 1) +2
2n + 2 , S(n) 2n +2n , n 3 2 2n
= 2n+1 , S(n+1) .
, 2n + 1 2n n 3.
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5.1.5
...},3,2,1{n μ
) )5(mod02)1(3 12 nn , . 5ν
) 6
)12)(1(...321 2222
nnnn ;
) 17 n 6ν
) nn 2 .
5.2
() 5.2.1.
2 2, . 4 4. 8
8. , 2n n N 2n.
3 3. λ λ. , 6λλλ3 λ 6+λ+λ+λ+3 λ. , 3n 3n.
5 5. 25 25. , 123...λ75 25, 75 25. , 5n n 5n.
7 7. , 203 7 20 – 2·3 ο 14, 14 7.
11 11. , 1 358 024 67λ 11 (3+8+2+6+λ) – (1+5+0+4+7) = 28 – 17 ο 11 11.
13 13. , 351 13 35 – λ·1 ο 26, 26 13.
. ђ, . , . 4 4.
, 57575746 4 14426 4,
57575736 4 12326 .
5.2.2. n .
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) 23 ν ) 47 3ν ) 56, n = 7; ) 98, n ο 12. , 10, 11, 12, ... A, B, C, … .
: ) 23 2 . , 2
. 21100123 . 10 . 23 : 2 = 12 1
12 : 2 = 6 06 : 2 = 3 0
3 : 2 = 1 1
1 : 2 = 0 1
211001
01234 2120202121 23100816
5.2.3. ( ). x, y Z y 0. q r x = qy + r, 0 r < |y| .
, 7, 3 Z 7 ο 23 + 1, 7 ,2 , 3 , 1 . , , .
. y > 0. je y 1. S = {x – uy | u Z, x – uy 0}. ђ (5.1.1) r S r =x – qy q Z. x = qy + r . .
S , 0 r , r < |y|. , r |y| = y . x – (q+1)y = (x – qy) – y = r – y 0. r – y S, r S . r |y| = -y , . , p r x = p|y| + r, 0 r < |y| , . x =py + r , q = - p x = qy + r, 0 r < |y| , ђ. .
q r .
q’ r’ x = qy + r = q’y + r’, 0 r , r ’ ξ еy|. r – r’ ο (q – q’)y . |r – r’е ο еq – q’ееy| . -|y| < r – r’ ξ еy|,
|r’ – r| < |y|. 0 |q – q’е ξ 1. q – q’ ,
, |r’ – r | ο 0. , q – q’ ο 0 r’ – r ο 0, . q = q’ r’ = r . q r .
5.2.4 x , ρ 0, q r x = qy + r, 0 r < y .
, q r x =
qy + r, 0 r < |y|. |y| = y , x = qy + r, 0 r < y .
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(nzd) d μ
1.
d | x d | y (d x d y, . x, y d);2. (cZ) c|x c|y c|d.
5.2.5 , . , d. ,
m, n d = mx + ny.
5.2.6 , nzd, d, m, n d = mx + ny:) , = 45, 126; ) 70, 154; ) 132, 156; ) 143, 165.
) nzd(45, 126) = 9 μ
126 = 2 45 + 3645 = 1 36 + 936 = 4 9 + 0,
9 = 45 - 136= 45 – 1 (126 – 2 45)= 3 45 + (-1) 126.
) d = 14, 70/154 2, 7ν
14 = m70 + n154 14, m = -2, n = 1; ) d = 12, m = 6, n =-5; ) 11, 7, -6.
, p 1 p.
, a b 1, . nzd(a, b) = 1.
5.2.7 a b .
a b m, n Z ma + nb = 1.
. a b . nzd(a, b) ο 1. 5.2.3 m n ma + nb = 1., ma + nb = 1 m n. d = nzd(a, b). d a b, . d|a d|b d|(ma + nb),
d|1, d = 1.
5.2.8 a, b, c Z a 0. , a|bc a, b , a|c.
. a, b , m, n Z ma + nb = 1. mac + nbc = c. a|bc a|a a|( mac + nbc) a|c.
5.2.9 a, b, p Z p . p|ab p|a p|b.
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. p|a . p a. p , p a .
p|b.
5.2.10 a1 , a2 , a3 , …, an , p Z p .
p| a 1a2a3…an p|ai i, 1 i n.
.
54 2 33 . , 496125 34 53 72 . .
5.2.11 ( ) k > 1
nc
n
ccpppk 21 21 ,
p1 , p2 ,…, pn , c1 , c2 ,…, cn .
5.2.12 () .
. p1 , p2 ,…, pn .
1...21 npppx . pi 1 x, i ο 1, 2, …, n.
, p p|x . , n + 1 . , ,
.
(n) n n
, . }1),(|{)( nmnzdnmmkardn .
, 1)2( , 2)3( , 2)4( , 4)12( .
5.2.13 1) n, 6 μ
) nn 3 ;
) nn 73 ;
) nn 113 .
2) n μ
) nn 5 5ν ) )67)(52)(1( nnn 6ν
) nn 7 7.
3) n A B:
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) 532 nnA , 2 nB ;
) 862 nnA , 3 nB ;
) 1582 nnA , 4 nB .
1) ) ) , , , 2, 3, 4, 5, . ν ) 6.
2) ) )(nf 5
k n 5 , 15 k n , 25 k n , 35 k n 45 k n , k ,
)1)(1)(1( 2 nnnn ν ) , 6
ν ) )1)(1)(1)(1( 22 nnnnnnn .
3) ) 1)2(54 22 nnn , ) 1)4(158 22 nnn .
5.3
5.3.1 a b n n
a – b )(mod nba „a b n“.
(1777 - 1855) 1801. 24. „DisquisitionesArithmeticae“, ,
.
, , , , ,, . Њ .
Ј μ , .
5.3.2
)(| ban , . n a – b nmba , mZ. ђμ )(mod nnmaa , mZ. a b n a b
n, )(mod nba )(| ban . μ )5(mod121364651 .
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5.3.3 1) μ
) )3(mod2834 , 34 – 28 = 23;) 7 ≡ - 1 (mod 4), 7 + 1 = 8 = 2 · 4;
) -19 ≡ - 4 (mod 5), jer je -19 – (-4) = (-3) · 5;) 23 ≡ 0 (ЦШН 23) , jer je 23 – 0 = 1 · 23.
2) a, b, c n μ
μ )(mod naa ;
μ )(modnba )(modnab ; μ )(modnba )(modncb )(modnca .
. a – a = 0 = 0 · n;. a – b= m · n b – a = (-m) · n;
. a – b= m1 · n b – c= m2 · n a – c= (m1 + m2) · n.
3) )(modnba )(modndc μ
1. )(mod ndbca ;
2. )(mod ndbca ;
3. )(mod ndbca ;
4. )(modnba mm , m ;
5. )(mod nkbka , k .
1. 2. a – b = m1 · n c – d = m2 · n (a ± c) – (b ± d) = (m1 ± m2)n;
3. (m1 m2) n2
= (a - b)( c - d) = ac – bc – ad + bd == ac – bd – bc + bd – ad + bd == ac – bd - b(c - d) - d(a - b) == ac – bd - (b m1)n - (d m2)n ac - bd = ((m1 m2) n + b m1 + d m2)n.
4. 5. 3.
, b = c = k.
4) 908019901980 37 10ς
4. 3. 7
2
= 49 = 50 – 1 -1 (mod 10) 74
=(72)2 (-1)2 1 (mod 10) , 32 = 9 = 10 – 1 (mod 10), . 34 1 (mod 10).
4|1980 4|80, )10(mod01137 908019901980 .
5) μ ) 3 13ν ) 11 13ν ) 6 7ν ) 3 11.
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36
) 3, 33 = 27 = 213 + 1, . )13(mod133 ,
3. 33 , )13(mod1399 , 3ν ) 1;
) 1ν ) 333 4025 (mod 11).
6. , ,, , . . , Q μ
}|{ ΝΖQ nmn
m.
, .
6.1.
.
6.1.1. μ ) 0,333... ) 0,272727...
) 3,121212... ) 12,0131313...
: ) ...333,0x 10
...333,310 x , xx 310 , 39 x , 9
3x ,
ν ) 100 3/11ν )
33
43 ; )
990
1312 .
Ј , , ... 5
1,5
2, ... .
, () (), () .
, . , , , , . , . .
,
2
11
2
12
2
3
,
4
21
4
6 . ,
( ) , .
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6.1.2. . ) . . ) , .
) 25 2. ) 25.
μ ) 2/5ν ) 7/15ν ) 13/11ν ) -3/4, -4/3.
6.1.3.
) 33
22
517
517
,
224
44
42025
45
,
4224
66
3355
35
.
) n:
i. 75
32
n
n
; ii. 314
421
n
n
; iii) 54
43
n
n
.
: ) 12/229; 1/9; 16; ) i. , a, b k ρ 1 2n + 3 = ak, 5n + 7 = bk. 5 2 1 = 5ak – 2bk, . k 5a – 2b , . ii.
314
171
n
n
17
12
11
n
. ,
, .
6.1.4.
) 10099
1...
43
1
32
1
21
1
; )
168421
16
1
8
1
4
1
2
1
1
1
1
aaaaaa
.
: ) 10099
99100...
32
23
21
12
10099
99
10099
100...
32
2
32
3
21
1
21
2
100
1
99
1...
3
1
2
1
2
1
1
1
100
99
100
1
1
1 ; )
32
1
32
a
,
.
, , () μ 1, 2, 3, ... . .
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, μ 0, 1, -1, 2, -2, 3, -3, ... . ђ , . , , (). – .
6.1.5. () .
: 2-. , . . .
6.2.
. , . μ
dc
ba , dcba :: .
, μ
cbda .
,21
14
3
2 143212 , . 42 ο 42.
6.2.1. μ
) 153
5 x ; )
3
2
5
7
x
x;
) 9
1
4
3
x
x
x
x; ) 3
6
7
5
x
x
x
x.
: ) 25; ) 31; ) 31; ) -27.
6.2.2. dcba :: μ
) ddcbba :: ; ) ddcbba :: ;) dcdcbaba :: ; ) dcdcbaba 54:3254:32 .
6.2.3. 5w
c
v
b
u
a μ
) wvu
cba
432
432
; ) 222
222
32
32
wvu
cba
.
: ) 5; ) 25.
wvucba ::::
, k w
c
v
b
u
a .
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6.2.4. :) 2:7:6:: yyx ;
) 5:11:15:)(:)( xxyxy ;
) 2:)(:)(10:: 2222 yxyxyxyx ;) 2:3:::33 yxyxyx .
: ) 63, 21; ) 33, 66; ) (0, 5), (5, 0)ν ) 2/7, 1/7.
6.2.5. μ
) 13:7: yx , 5:7: zy ; ) 4:9: nm , 5:6: pm ;
) 5:3:2)(:)(:)( baacbc ; ) 7:10:11)(:)(:)( wvwuvu .
: ) 35:92:49:: zyx ; ) 15:8:18:: pnm ; ) 5:3:2:: cba ; )
3:4:7:: wvu .
nGGm :: mnG , ,
. ,
2
nmA
nm
H11
2
m
n .
6.2.6. m n.) 4 6.
) . ) G, A H
m n AHG .
) AGH ,
nm .
: ) 4 λ 946 . ) ,
)(:)(: kbkaba b
a
kb
ka
b
a , . . ) Ј
mnnm
mnnmAH
22
. ) 02
nm .
, , .3 abc .
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40
. , ACABABAC :: ,
yxxyx ::)( .
1y , . 1 xAC
1::)1( xxx , 12
xx .
12 xx , ¼ 4
5
4
12 xx ,
4
5
2
1 2
x .
2
51x .
. 2
51
1λ0λ. 450 ....
6.2.7. …
287498948481.61803398 .
)
...1
11
11
11
.
) ...111 .
: )
...111
11
11
x .
xx
11 . „“ 12 xx . )
...111 x . ,
xx 12 , . „“ .
6.3.
ђ . , . .
. , , , .
6.3.1. . ) 126 2 20 . λ0 ς
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) 6 . 7 ς
: ) , 126 140 , , λ0 .
ђ , . ђ .
μ 140 ο λ0 μ 126. ο 100 , . 1 40 .) , .
, . : 6 = 4 : 7,
7
33x , . 3 26 .
6.3.2. ) 14 17 . 5 6 . ς ) 6 5 . 2 ђ 3 ς
: ) 14 12 , 8 . , . μ 12 ο 14 μ 8,
12
8
14x , . 21 .
) , 1 6 5 , 3/5 λ .
, , . , .
: 5 = 6 : 9ο 3/5 μ 1 ,
μ 5 , ,
, , . 19
)5 / 3(6
5:
x . ο 2 .
6.3.3. ) . ς
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) 6 , 8 λ 18 . 7 , 6 8ς ) 5 , 6 , 8 140 . 6 , 5 , 7
ς
: . ) 1,5 1,5 1,5 , , . 1,5 3 3 . , , . 3 3 6 . , 6 , . 1 3 2 ., , . 1 1 2/3 .
) 9861
V (
), 8672 V .
xVV :18: 21 , 18986
867 x , .
14x . ) , , , . .
x : 140 = 6 : 5= 5 : 6
= 7 : 8
865
756140:
x , ,
. , ο 160 .
6.3.4. . ) 500 2, 3 5. ) Ј ђ, 3 (ђ) 2% , 1,5% 1,5 % , . ς ) 800 20% .
: ) x, y z. 500 zyx
5:3:2:: zyx .
k x 2, k y 3
k z 5
, 500532 k k k , 50k . 100x , 150y 250z
. ) ђ λ5% 3000, . 2850 , 60 , 45 , 45 . ) 220, 264 316 .
6.3.5.
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) Ј 700 . 6 7 , 5 8 , 3 12. ς ) 800 , 400, 500 600 . ς
) λ00 40%, 30% 30% , 7 , 11 , 2 .
: ) , 42, 40 36, 118 . , 42/118, 40/118 36/118 700 , . 24λ, 237 214 , . ) 4/15, 5/15 6/15 800 , . 213, 267 320 . ) . , . 2/11 λ00 , . 164 .
40% μ 30% μ 30%, 66, 4λ 4λ . 5 . 5/λ λ00 40% μ 30%, .
2347
4900
9
5 , 175
7
3900
9
5 .
4 , 4/11 λ00, . 327 . , , 66 + 234, 4λ + 175 + 327 4λ , . 300, 551 4λ , .
6.3.6. . ) ђ 200 1,1 300 1,2 . ς
) 60 30 λ0 50 ς ) 65 %, 60 % 45 % λ00 50 %ς
: ) 500 , x 5002,13001,1200 . , ο 1,16 . ) Ј 90 yx 90503060 yx , = 60, ο 30 .
50 60 30.) , 50 % 50, 100 150
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. x, y z .
900 zyx
150:100:50:: zyx 150, 300
450 , .
.
6.3.7. . ) λλ% . 98% . ς ) 1λ8 2 . λ8%.
: ) , v s .
100
99
sv
v. r
rv , λ8%. 100
98
srv
rv . r ο 0,4λ5
4λ,5% , 50,5% . ) λλ% . 1 % , 2%. 2 2% , 100 . λ8 . , 100 .
100
pTGK , K , G
, T , p .
365 ( 366) , 360 30 . , 7λ
0,216365
79T , 0,219
360
79T .
6.4.
(, ) .
. , . , .
6.4.1. . ) .
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) . .
) n N .
: ) , , , , . ) 24 . ) nn ...21! , .
„ “.
6.4.2. . ) () . ) () . ) . ) k- n- ς
: ) , , , , , . ) k n . k n . n , n .
n-1 , n(n-1) . , , - n-k +1 ,
)1(...)1( k nnnVnk .
6.4.3. (). ) 0-5?) 0-4?
: ) , 0, 1, ..., 5,
. , , 5 . , , 5 .
4 . 554 = 100. ) 4432 = 96.
6.4.4. . ) 0-5?) 0-4?
: ) , .
1-5. , 0-5. 566 ο 120 . ) 500.
6.4.5. . ) 2- 5 ς ) 3- 5 ς ) k- n ς ) 10- μ 2-, 8-ς
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: ) (, ) 5- . 5
, 4, 54 . , , .
, 2! ο 12. , 2 5 10
21
4552
C . )
10321
34553
C . )
k
k nnn
k
VC
n
k n
k
...21
)1...()1(
!. ) , 45,
10810
2)8...3(21
)3...8(910
21
910CC
.
7.
7.1. 3
34 . „()
“, „ “. ( ) ,
. (1,732)2 3 3 1,732 3 4 1,587 (1,587)3 4.
μ 2 = 1,414... ; 3 = 1,732... ; 4 = 2 ; 5 ο 2,236... ν .
. ( 6.1.1.) . .
7.1.1. 2 .
: , n
m2 , m n
. Ј . ђ,
22 2nm .
, m , m = 2k, k = 1, 2, 3, ... , k ( m).
, m 22 2)2( nk , . 22 2k n .
n . , m n , .
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, ђ, 2
.
μ
yxxy y
x
y
x ,
. ђ,
xx
x
x
x
x
1.
7.1.2. μ
) 4520 ; ) 156 ; ) )234)(234( ; ) 18
23.
: ) 5954 5352 55 ; ) 5332 592
109 103 ; ) 22 )2()34( 44448 ; ) 29
23 1
23
23 .
7.1.3.
μ ) 13
1
; ) 6223
34
.
: )
13
13
13
1
2
13 ; )
2362
2362
2362
34
1824
612188
6224 .
7.1.4. ABC CD. AC = 15 cm CD = 10 cm. AD = x, AB = y BC = z.
: 222 1015 x ,
. 55x cm. ACD ABC
15 μ x = y : 15 = z : 10. 59y cm
56z cm.
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7.2.
, ( ) () . a, b c
(a + b) + c = a + (b + c), (a · b) · c = a · (b · c).
, (2 + 3) + 5 ο 5 + 5 ο 10 2 + (3 + 5) ο 2 + 8 ο 10. , (2 · 3) · 5 = 6 · 5 ο 30 ђ 2 · (3 · 5) = 2 · 15 = 30.
, . acabcba )(
. ( ) ( ). 2 · (3 + 5) = 2 · 3 + 2 · 5, 16.
, .
7.2.1. μ
(a + b)2 = a
2 + 2ab + b
2 , ν
(a + b)3 = a
3 + 3a
2b + 3ab
2 + b
3 , ν
(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 , .
7.2.2. b – b μ
(a – b)2 = a2 – 2ab + b2 , ν
(a – b)3
= a3
– 3a2
b + 3ab2
– b3
, ν (a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4 , .
7.2.3. μ
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
( ) μ
(a ± b)5 = a
5 ± 5a
4b + 10a
3b
2 ± 10a
2b
3 + 5ab
4 ± b
5.
7.2.4. μ
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a2 – b2 = (a – b)(a + b), ν
a3 – b3 = (a – b)(a2 + ab + b2), ν
a3 + b3 = (a + b)(a2 – ab + b2), .
7.2.5. , μ
) x
5
+ x + 1; ) x
4
+ x
2
y
2
+ y
4
; ) (x + y + z)
3
– x
3
– y
3
– z
3
.
: ) 1225 xxxx )1()1( 232 xxxx
)1)(1( 232 xxxx ; ) 224224 2 yxyyxx 2222 )()( xyyx
))(( 2222 xyyxxyyx ; ) )())(( 3333 zyxzyx
))(())())(()(( 2222 zyzyzyxzyxxzyxxzyx
))()2333)((( 22222 zyzyyzxzxyzyxzy
)3333)(( 2 yzxzxyxyx ))()()((3 yxzyxxzy
))()((3 zyzxyx .
7.2.6. 1. μ
) 12 2 xx ν ) 123 2 aa ν ) 145 2 bb ν ) 167 2 cc .2. μ
) 22 5136 baba ν ) 22 31914 vuvu ν ) 22 7910 yxyx .
3. μ
) 32 a ν ) 253 x ν ) vu 23 ν ) 2332 .
4. μ
) ...124 2 xx ν ) ...2309 2 aa ν ) ...129 2 mnm ν ) ...61212 .
5. μ bcacabcbacba 222)(
2222 .
6. μ
) 12 a ν ) 3x ν ) 23 ν ) b52 .
7. μ
) ...128 23 xx ν ) ...9 23 aa ν ) ...2933 ν ) ...36324 c .8. :
) 22)( cba ν )4
1)
2
15( 2 ν )
9
4
5
12
2
x ν ) 213 2 .
9. , μ
) 125)1( 3 a ν ) 343)3( x ν ) 16 x ν ) 13333 baba .
10. μ
) 133 1255 nn aa ν ) 124 813 nn bb ν ) 3284 235 nnn ccc ν ) 51256 36 nn xx .
7.3.
, , .
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. .
7.3.1. ) 5
3
75
73
35
21
; )
12
2
x
xx
)1)(1(
)1(
xx
xx
1x
x, 01x .
. () , () .
7.3.2. ) 7
4
5
3
3
2
105
60
105
63
105
70
105
67; )
1
3
1
21
xxx
)1)(1(
)1(3
)1)(1(
)1(2
)1)(1(
)1)(1(
xxx
xx
xxx
xx
xxx
xx
)1(
)(3)(212
222
xx
xxxxx
xx
xx
3
2 152 .
() () . μ
bd
ac
d
c
b
a ,
db
ca
d
c
b
a
:
:: .
, .
7.3.3. ) 35
6
7
2
5
3 ,
5
2
7
5:
35
10 ; )
111 2
2
x
x
x
x
x
x,
1
1
1
1:
1
123
2
xx
x
x
x
x
x, 1x .
. μ
c
d
b
a
d
c
b
a
: .
7.3.4. ) 13
9:
10
7
90
91
9
13
10
7 ; )
1:
1 x
x
x
x
x
x
x
x 1
1
)1(
)1(
xx
xx
( )1
1
x
x, x 0.
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, ( ) , ( ) μ
bc
ad
dc
b
a
d
c
b
a: .
7.3.5. μ
)
aa
a
a
1
1
1
; )
21
11
1
11
b
bb
; )
13
2
9
4
127
8
2
2
3
3
b
a
b
a
b
a
; )
n
n
bbbb
b
1...
111
1
32
.
: )
1
1
2
a
aa
a aa
aa
2
1
3
2
aa
aa
2
13
3
24
, 1,1,0 a ; ) 1 – b, b
±1; ) 13
2
b
a, b 0; )
)1...(1
)...1)(1(
321
12
nnn
n
n
bbbb
bbbb)1( bb
n , b 0.
7.3.6. μ
1. ) 23
2
2
2
1
12
xx
x
x
x
x
x; )
65
7
3
2
2
32
aa
a
a
a
a
a .
2. )
nm
nm
nm
m
m
n
n
m1
22 ; )
x
y
y
x
yxyxx
yx 11112
2
.
3. )
u
v
v
u
vu
v
vuuvu
u2:
22
; )
nm
nm
nm
nm
nm
nm
nm
nm: .
4. )
bc
cba
bca
c
acbca
abcba
21:
2
2
2 222
222
222
;
)
vw
uwv
wvuwvu 21:
11:
11 222.
8.
1. n:) nnn 45 33 120ν
) nnn 23 23 6.
2. n :
) )67)(52)(1( nnn 6ν
) 12 n 3.
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3. 7 , λ. ς
4. 7
λ 2μ3. ς
5. n ,
) 3322 5454 nnnn 10ν
) nnnn 21122 3232 6.
6. , m )(modmyx ,
n )(mod myx nn .
7. μ
11 f , 1
2 f , n
nnn
f f f
12
.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610,... . n:
) n
f 3
ν
) n
f 4
3ν
) k nk f f nk || .8. . . ς
9. . 42. ς
10. , 78% , 20% , 1% . 1000 ς
11. 2007. 2410, 7187, 4211 37. ς
12. 450 5 λ0 . 7 75 ς .
13. 5 7 115 . 3 3 ς
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14. 5,5 6,25 . λ00 6 ς
15. 10,10 , λ,65 .
120 λ,80 ς
16. , 27% 370, 40% 250ς
17. . , . , ς
18. 5000 8% . , ς , .
ς
19. 7000 6% 5 . , ς ς
20. 3 .
21. 32 .
22. μ
) )234)(234()234( 2
ν ) )81(2)122(2
1
.
23. 20 cm 12 cm.
μ
24. ) 9)64)(4( 22 xxxx ν ) 16)85)(5( 22 yyyy ;
) 24)105)(5( 22 aaaa ν ) 120)227)(7( 22 bbbb .
25. ) 14610 xxx ν ) 12 23456 uuuuuu ;
) 11664 36 yy ν ) 12 23456
vvvvvv .
26. ) )()()( 222 bacacbcba ;
) )()()( 222 bacacbcba ;
) 333 )()()( accbba .
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27. μ
) ))(()()( 222222 yxbabxaybyax ;
) ))(()()( 222222 yxbabxaybyax ;
) ))(()()( 222222 yxbabxaybyax ;
)
222222
)()())(( aybxbyaxbayx .
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III . , , .
.
p P . P .
ν , . , .
p ђ A B,
. A B . T p p AB .
AB , , () , . , , , .
. , . .
.
. , . .
p . p .
.
, , , , , .
, ,
p. p p , p . , . ()
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. , .
. ,
. .
ђ 6. ... . (.), 6. 3. ... .
13 џ , , .
, 17. , , . , . 18. . , 1λ. ,
, , , , .
1λ. μ , (David Hilbert, Grundlagen der Geometrie,18λλ), . ђ ђ, . 1λ57. , , . 8. . .
9. А ђ.
.
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. , , .
Ј ђ ,
.
. 1λ. μ
1. (1-8);
2. (1-4);3. (1-5);4.
ν 5.
(1-2).
9.1. ,
ђ , . .
I-1 . I-2 .I-3 . . I-4 , . . I-5 .
I-6 , .I-7 , .I-8 .
A, B, C,...; , , a, b, c, ...ν , ,
, , , ... ν .
I-7 , I-8 . I-1-3 I I-4-8 I.
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(I-1,2), A B a = b .
, , , ... ,, , , .
9.1.1. a b .
: , . , a b , A B. I-1,2 . , , . a = b. ,
, ђ .
. I-3 ђ .
9.1.2. ђ .
I-4.
ђ .
9.1.3. ђ .
: p C , . I-3, p A B, I-4, A, B C
. A B p
, I-6 . , p C A, B C,
I-4,5 . , ђ p
C.
9.1.4. ђ .
: }{Aqp , . I-3 p
A B, q A C.
I-4 A, B C . I-5
.
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9.1.5. ν .
: А, I-7
B. А B I-1,2 p I-6 , А B ., p.
, p , p , I-5 .
, .
9.1.6.
.
: , I-6 , . . ,
.
. I-8 ђ .
9.1.7.
1. . ς
2. . ς
3. ђ ς
4. ђ 5 4 ς
5. ђ ς
9.2.
"ђ" , .
II-1 B ђ A C, A, B, C B ђ ђ C А.
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II-2 А C B AB, C ђ A B.II-3 ђ . A, B C ,
B ђ A C. II-1 B ђ C A.
ђ ђ .
II-4 A, B C a ABC ν AB, AC, BC.
. 1
. II-4 , μ .
ABC , , BC. AC BC.
9.3. , .
III-1 A, B , , А' ', B' ' А', AB А'B', μ AB А'B'.III-2 А'B' А"B" АB, А'B' А"B", μ , ђ .
III-3 АB BC ,, А'B' B'C' ' ν AB А'B' BC B'C', AC А'C'.
1 Moritz Pasch: Vorlesungen uber neuere Geometrie, Leipzig 1882.
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, AB BA. , . , . .
(a, b) (b, a)
b . Oa Ob , . , .
ђ
(), . O
A B AOB, B. .
III-4 (p, q) α a' α' ђ
α' a '. p' a ' 'ν α' q' (p, q) (p', q') (p',q') К', (p, q) (p', q'). .
9.4.
9.4.1. , .
9.4.2. .
IV ( ). А μ , ђ А, А .
9.4.3. ђ .
: IV, a, b .
, ,
, I-5 .
p p . p .
b А a a, А a.
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b . , ђ .
, , . (12.4.1.) ,
(12.4.2) . 12.1.4. , . .
9.4.4. .
9.5.
V-1 ( ). AB CD , n, CD n A B B.V-2 ( ).
ђ I-III V-1.
10. , , . ђ . , .
, a b t . A
B a b. 1, 2, ..., 4.
AB, 1, 2, 3 4.
, 3, 4, 1 2.
,
3 3.
, 3 1.
, 1 2.
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10.1.
,
180 (). 10.1.1. .
180 . ( ) . , . , . . 10.1.2.
180. 10.1.3. .
10.1.4. 360.
10.1.5. μ
) 360.) 540.) n- 180)2(n .
) n- 360.
: ) n- n д3, 4, 5, …ж. А1 n-3 , A1A3, A1A4, A1A5, …, A1An-1.
n-2 А1 А2 А3, А1 А3 А4, А1 А4 А5, ..., А1 Аn-1 Аn. 180)2(n .
n-2 А1 . ђ, , Аn.
)
. 1, 2, 3, ..., n, 1, 2, 3,
..., n, . 1 + 1 = 180, 2 + 2 = 180, 3 + 3 = 180, ... , n + n = 180. 1+ 2+ 3+...+ n= (180- 1) +(180- 2) + (180- 3) + ... + (180- n) = n180 + 1+ 2+ 3+...+ n = n180 + (n-2)180 = 360.
Ј
, 60.
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Ј . ().
.
10.1.6. .
: ABC, BC > AC, . a > b. BC D, C-D-B CD = b. ADC , A D
, . BAC =
ABD, = ABD, . > >
, > , .
10.1.7. .
, > , a > b.
b > a.
( ) . . 2:1 .
10.2.
10.2.1. CD ABC:
1. 2
ABCD C > 90.
2. 2
ABCD C = 90.
3. 2
ABCD C < 90.
: 1. ABC . AD > CD ACD > , DB > CD DCB > . > + , 180, > 180 - , > 90. 2. AD CD ( DB CD) , .
ACD = ( DCB > ). = + , = 90.
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λ0 ( ) , .
λ0, . ,
λ0. . .
10.2.2. .
: C B D. ABCD ,
360, + 2 = 180. ђ, 1 + 2 = 180.
10.2.3. ABC 3645'30'' 6225'30'' .) , C ς ) . ђ . ) B.
: ) , , 60. ,) , ђ .
. .
10.2.4.
135.: ABC
A B . , . + = 90. ABS,
, /2 /2, 45. , S 135. , S 135, B () 45, λ0. C.
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10.2.5. Ј 5 - 12 -. .
: , Ј
ABC.ABC = = 90
222
125 x , . ο 13 .
Ј ,
45 + = 112,38.
10.3.
( ) , . Ј , .
. .
10.3.1. . . 1. 2. 3.
1. AOB BOC,
. OBC = , = 2 . 2. , AD DB, 1 = 2 1 2= 2 2 ,
= 1 + 2 = 2 . 3. = 1 – 2,
AD BD.
10.3.2. ( ) .
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) . ) (), () . ) ђ .
: ) 2. CAD , ACD . BCD.
180. ) , tAB ACB .
10.4.
1. . .
2. .
3.
λ. .
4. ς
5. ђ .
6. ABC CD AB 105. CD = BC.
7. AA’ BB’ ABC AB
C.
8. H ( ha hb) ABC,
AHB +C = 180.
11.
, ( )
, , . , .
ђ , , , , ,
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, . ђ, , ( )μ
) ν )
ν ) ν ) , .
() .
AB1C AB2C , (),
b.
1 2 1 2.
11.0.1. Opq M. P М Op Q M Oq MP = MQ, OM Opq.
: , OMP OMQ
(P = Q = 90), , OM.
MP = MQ, OMP OMQ, POM = QOM.
11.1.
11.1.1. ђ .
: , ABC A1B1C1 μ
11CBBC , 11CAAC
11DCCD
, . a a1, b b1 h h1. , .D = 90. a b.
,
Q ADC A1D1C1 . D = D1, a = a1 h = h1, ,
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( 10.1.7.), . b h,
ADC A1D1C1. A = A1.
, a = a1 , b = b1 a b A = A1,
ABC A1B1C1.
, ( ) . . () 2:1 .
11.1.2. , . .
: ABC, ,
CD. = =C/2, AD = DB.
, CD
E C-D-E CD = DE. BDC =ADE ( ), BDC ADE. BC =AE = . AEC , , . AC = AE. AC = BC, .
ABC .
11.1.3. .
: AC = BC , EC = CD = , . C
AEC BDC, AEC BDC, AE = BD, . .
, , . AE = BD, AT= BT ο 2/3 , DT = ET ο 1/3 ,
ATD = BTE. ATD BTE. AD = BE, AC = BC, ABC .
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11.1.4. ( , 17. 1λ83. ) ABC A1B1C1 . D D1
AB A1B1. BCD =B1C1D1, AD = A1D1.
: BCD B1C1D1
( ), DB = D1B1 AB – DB = A1B1 – D1B1, AD =A1D1.
11.1.5.
1. ABC A’ B’ C’ μ ) b = b’ , ha = ha’ , = ’ ; ) hc = hc’ , = ’ , = ’ ;) a = a’ , c = c’ , tc = tc’ ; ) hc = hc’ , tc = tc’ , c = c’ .
2. μ ) ν ) .
3. .
4. M AB AM = BM. .
5. . μ ) ν ) .
6. , ( ) ( ) .
11.2.
, , , . , .
11.2.1. .
,
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. , ABCD AC.
ACB =CAD, CAB =ACD, ABC CDA.
. .
11.2.2. .
, AB CD BC DA ABCD, .
E ABE CDE ( )
. , , ( ). .
11.2.3. .
11.2.4. .
: ABC M N AC BC, . MN
. MN P P-M-N PM = MN. (3) ANCP . (2) CN PA . , (2)
ABNP . , MN AB AB = 2MN.
11.2.5. M N AB CD ABCD. DM BN AC .
: , DM BN E F.
AME CNF () AE= FC. EG || MB MBGE
EGF AME (), ђ AE = EF. DM BN
AC .
11.2.6. 2μ1 .
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: ABC () AS CM , . AS D. ABDC.
, CM BN (N
CD) AD T V AT = TV = VD. S
TV AT = 2TS, T AS 2μ1 A.
11.2.7. . . .
,
.
.
11.2.8. ( , 15. 1λ84.) ABCD AB > BC. B1 B AC, E AB1 CD. ADE CB1E.
: , CAB DCA . CAB CAB1
AB1 AB AC., DCA CAB1, AEC (AE = EC). , CB1 = CB = DA ђ DEA CEB1,
ADE CB1E .
11.2.9. DC ABCD M. BAM BC N. AM = DM + BN.
: DC P P-D-M PD = BN, .
ADP ABN, APD PAM, APD = ANB= 90 - , PAM =PAD + DAM = + (90 -2 ) = 90 - . , AMP . AM = MP = DM + DP
= MD + BN.
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. , , . , . Ј
, – .
. .
11.2.10.1. μ ) ν ) ν ) ν ) .
2. ђ .
3. .
4. AC ABCD A
C. ABC ACD.
5. , .
6. M N AD BC ABCD, AB
+ CD = 2MN.
7. , .
8. ABCD ( ) . AB CD BC AD, M N AC BD AC BD.
11.3.
10.3.
( 10.3.1.). () ( 10.3.2.) , ђ.
() . () .
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11.3.1. ABCD AB +CD = BC + AD. еAB – CD| = |AD – BC|.
.
ABCD M, N, P Q, AB, BC,CD DA. A, B, C D , . AM = AQ, BN = BM, CN = CP DP =
DQ. AB + CD = BC + AD.
11.3.2.
(180).
: ABCD. ( ) ( A C)
BD, . + = 180.
11.3.3. A.
B C. BAC (λ0).
: , AS A SA SB , SA SB . SA = SB = SC. , S A, B C, A
BC .
11.3.4.
1. 65, 105 115. ς
2. A, B C 1μ2μ3. ABC.
3. s – a, s – b
s – c, 2
cbas
.
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4. . .
5. . .
6. А BC D. ABC ABD ς
[30]7. O AB. A B C D. COD .
11.4.
(&10.). ђ,
180 ( 10.1.2.), ( 10.1.3.),
360 ( 10.1.4.). ђ, .
, , .
11.4.1. .
: ABC AO BO sA sB. OM,
ON OP BC,CA AB.
APO ANO
/2. OP = ON. ђ,
BPO BMO
/2, OP = OM. OP = ON = OM.
CMO CNO, , OM = ON.
OCN OCM, CO , . OP = ON = OM
.
11.4.2. .
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: ABC S sa sb BC CA. M N .
MBS MCS ,
. BS = CS. NCS NAS. CS = AS. AS = BS = CS.
, ABS S sc AB. ђ, S
ABC.
.
11.4.3. , .
: , A, B C ABC NP, PM MN BC, CA AB.
BMC, CNA APB ,
, . A, B C NP, PM MN.
(11.1.2.), MNP
H, ABC.
, . . (&11.1.). ( 11.2.6) 2:1 .
ђ, , , . ( 11.2.4.) .
11.4.4.
1. 2μ1 . 2. , . .
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. 1. , AN BM ABC T. MN , AB .
ABT PQ, ђ AB . MN || PQ MN = PQ. MNT PQT
MNT PQT, PT = TN QT = TM. P AT Q BT, 2μ1 A B.2. AN BM ABC T.
CP BM S , .
, BS = 2SM, BMBS
3
2 . , BT = 2TM
BMBT3
2 . BS = BT,
S T
. , CP , .
11.4.5.
1. ђ .
2. .
3. .
4. D AB ABC. E
BC DE BC. F DE. AE CF.
[ CD . G BE GE
BED GF || BD, GF CD. F CDG, DE GF. , CF CF DG. ђ, DG
ABE AE || DG AE CF.]
11.5.
(10.1.6. 7.).
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,.
11.5.1. ( ) , .
: BC ABC D-B-C DC = CA = b, . ACD
DAC =ADC = . BAC = , < BAD, BDA ’ c > c’.
: A =A’ C = C’, CB’
. s BCB’ AB D. DCB DCB’ ( ), DB = DB’., AD + DB’ ρ AB’,
AB > AB’.
11.5.3. , a = a’, b = b’ c > c’ > ’.
: C C’ μ = ’, < ’, > ’. 1) = ’, ABC A’ B’ C’ , c = c’, . 2) < ’ 11.2.2. 3) > ’ .
11.5.4.
1. ђ .
2. .
3. .
[ O ABC . , ABO: /2 /2 BO AO, BOC: /2 < /2 CO < BO. , CO.]
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4. .
5. ђ .
6. .
[ tc < b + c/2 tc < a + c/2]
7. , . .
8. ¾ .
9. μ) ν) .
10. ABC M. :
) ACB
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IV .
, , , (&13), (). ђ ђ (&14.).
, (μ "υ ῖ " – ) , . , , , , ђ.
12. Кј 2.
, .
. :
1μ ν 2: ν 3μ ν 4μ ν
5μ ν 6μ .
:
ν
(--) ν
(--) ν
(--) ђν (--) .
, , . μђ, ђ, , , , , , .
2 (42λ-384. ....)μ „
.“
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μ
, ν
, ν
,
ν , ђ
.
. . μ
. , .
. , .
. , .
12.1.
12.1.1. a, b ha, .
. () ABC. a, B C . A C b, BC ha.
. , k(B,a) B a Bx BC.
.
BC : Bx; C = Bx k(B,a). k 1(C,b).
n BC C; P, Q n k 2(C, ha). p q P Q BC.
А: p k 1(C,b), q k 1(C,b), . AiBC (i = 1, 2, 3, 4 ), .
.
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