. ij, *. £ * * £ . = *. I I*. .
, Z , , N .
, R={a,b,c,d,e,f} a8 = {(a,a), (b,b), (c,c), (a,b), (b,c), (a,c), (d,d), (e,e), (f,f), (d,e), (e,f), (d,f)}. a, d c, f .
. , - , £ £ , .
£ ' :
< y, , £ , ¹ (, , );
, , £ , £ .
(,) Î R, R , : = , < , > , .
? ' , ' , . , , . . , .
m R (), ri Î R, ri < m (ri > m).
, ( I), () .
, : R={a, b, c, d, e, f} a8 = {(a,a), (b,b), (c,c), (d,d), (e,e), (f,f), (a,b), (b,c), (a,c), (d,e), (e,f), (d,f)}. a d , c f - .
, .
2. R={r1,...,rn} : R={1 ...n }, xi < xj i < j.
. . : , £ . , . - xi < xj xi . .
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R S .
. Î R () S, £ x (a ³ x) ÎS.
. b Î R S, :
1) b S;
2) b ³ b¢, b¢ - - S.
. c Î R S, :
1) c S;
2) c £ c¢, c¢ - - S.
b = inf S, c = sup S (infimum i, supremum e). " " " ", .
, .
, ᒺ , , , , . , .
. , N, Z, Q R. Z. , ᒺ Z : Z = NÈ{0}ÈA.
A = {-x: x Î N}. , Z, x {NÈ{0}ÈA}.
x = , x£ £ x. x ¹ . :
) x, Î N, Z , N;
) x, Î , x £ , - £ -x N ( - 5 £ -4, 4 £ 5);
) x = 0 Î N, x £ y;
) x Î = 0, x £ y;
) x Î yÎN, x £ y £ x.
Z Q:
a/b£ c/d , ×d < b×.
, . , R. :
D = 0dnd2d1d0d1d2,
C = 0cmc2c1c0g1g2.
di = ci di = gi i, D = C , , D £ C C£ D. :
) dn ¹ 0, cm ¹ 0 n ¹ m, D £ C, n < m, C£ D, m < n;
) n = m di ¹ ci, dj = cj j , i < j £ n, di < ci , D £ C, , , ci < di, C£ D;
) n = m di = ci i, dk ¹ gk k dj = gj j , 0 < j £ k, C£ D, gk < dk, D £ C, dk < gk.
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. ³䒺 , Z.
, , R , :
[, b] = {x: x Î R, £ x £ b} - () b;
], b[ = {x: x Î R, < x < b} - b.
b . , . :
[a, b[ = {x: xÎ R, a £ x < b},
]a, b] = {x: xÎR, a < x £ b},
:
] - ¥, ] = {x: x £ },
]-¥, [ = {x: x < },
[, ¥[ = {x: £ x},
]a,¥[ = {x: a < x},
]-¥,¥[ = R.
.
2. ³ . .
. R, , , E =.
.
, . , .