15
1 35. , , , , (!) ?
1.
2.
3.
4.
5.
6. 2- R.
7.
8.
9.
10.
11.
12. , n.
13.
14.
15. n - R.
16. .
17.
18.
19.
20.
21.
22.
23. , 3.
24. 3- R.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
.
1. Ê = (; , ) (2,2) , :
I. (; ) ;
II. (; ) ;
III. .
I - III , , .
, , :
I. 1. ( , b, c )[( b) = (b )].
2. ( )( )[ = = ].
3. (" )( s () )[ s () = s () = ].
4. ( , b )[ b = b ].
II. 5. ( , b, )[( b) = (b )].
III. 6. ( , b, )[( b) =( ) (b ) ^ (b ) = ( b) ( )].
,
III. 7. ( , b )[ b = b ],
Ê .
, . , , , ; , , Ê, , .
, 5 , - . . .
Ê , 1Ê.
1Ê , .
Ê (), ().
Ê , = {0}.
Ê, , .
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, Ê , .. +, , .. ×. 1 6 Ê = (; +, ×) :
I. 1. ( , b, )[ (a + b)+ c= + (b + ) ].
2. ( 0 )( )[0 + = + 0 = ].
3. ( )( (- ) )[ + (- ) = (-) + = 0 ].
4. ( , b )[ + b = b + ].
II. 5. ( , b, )[ ( . b) . c = a . (b . c) ].
III. 6. ( , b, )[ ( + b) . c = a . + b . c ^ . (b + c) =
= . b + a . ].
.
2. (; +) Ê= (; +, ×). Ê 0Ê 0 ( ).
3. Ê= (; +, ×) - 1Ê, (; 1k) Ê. 1 .
4. Ê , , ..
V. 7. ( , b )[ × b = b × ].
. N, Z, Q, R, C, 1, 4, 5, 6, 7 , 2, 3.
1 10
1. (N; +, ×),(Z; +, ×) , . . 2, 3.
2. Z = (Z; +, ×) - .
3. Z = (Z; +, ×) - .
4. Q = (Q; +, ×) - .
5. R = (R; +, ×)- .
6. Q [ ] = { q 1 + q 2 | q 1, q 2 Q },
Q [ ] = (Q [ ]; +, ×).- 1. 1 7. 1Q = 1+ . 0, q 1+ q 2 (- q 1) + (- q 2).
7. 0= ({ 0 }; +, ×) - .
8. R[ x ]= (R[ x ]; +, ×) - x .
9. . Z , m , 1, Z p
( z1, z2 Z)[ z 1r z 2< = > (a b) m ] (1)
, r . , Z
= { mk | k Z}, ={ mk + 1 | k Z }, , ={ mk +(m +1) | k Z},
m. Zm ={ ; ; ; }.
.
( , Zm) [ ],
( , Zm) [ ]
(- )= , = , , = mq + r, 0 r < m.
, Z m m - Z= (Z; +; ×) m.
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0Ê , , m.
- .
1k; = .
, Zm= (Zm; +, ×) - ( m) .
10. 2- (3-) (M 2 2; +, ×).
,
Ê = (;+, ) , + . . . , , (R;+, ) , 0; (Z;+, ) .
, . .
1. , , = =1 . .
2. P =(P;+, ), , .
, :
I. ;+, ) ;
II. (; ) ;
III. +;
IV. ;
V. .
I V, , :
I. 1.
P2.
P3.
P4.
II. P5.
III. P6.
IV. P7.
V. P8.
1 8 .
3. (;+) Ð. Q ( ).
, , . 2,3,7, 1,4,5,6,8 .
1- 5
1. (N;+,×) , 2, 3, 7.
2. (Q;+,×) , 0 ( 7 ).
3. (Q*=Q\ ;+,×) .
4. (R;+,×) , 7, (R=R*\{0};+,×) .
5. , Q [ ]=(Q[ ]= ;+,×) . Q[ ].
q1+q1/, q2+q2/, q1q2/+3q2q2/, q1q2/+q2q1/ . , Q[ ] .
2, 3, 7.
2. Q 0+0 , 2 .
3. 3 a+=a/ :
:
, 1=q1/-q1 2=q2/-q2 . , Q[ ] , 3 .
7. Q[ ]
a×x=a (*)
a¹. (*),
, 1+ 2, 1,2 , (*) Q[ ].
, . , , q1 ,q2 , , . , q1¹,q2¹. , , 3 , . , , . , (*) Q [ ] Q [ ] .
6. , , , . + × . , . 1,2,4,6, . , 3 8:
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, (;+,×) .
, (;+,×) ,
= (**),
, . , . . A¹0, , . . , , , , . : , . , , = .
, (;+,×) .