.
.
, .
. , , , .
, , .. :
= {, , , }
B = {k, m, n}
C = {5; -7; 0,9; 100; 8}
D - 5 10
E = {}
F - 7 100
G = {, }
N
..
(), (A, B, C, E, G), (D, F, N) , .. . Æ.
, . Î ( , Ï). ,
Î ; 5 Î ; 10Ï .
, Ì. , , .. Ì .
, .
, , . È. , DÈF - 5 100.
, , . Ç. , DÇF - 7 10; CÇD = {5; 8}; Ç = {}; ÇG = Æ.
, , , . \. , {, , }; \D = {-7; 0,9; 100}.
, , , , . , Ì {, , } =\.
, [1] , . , C, D, F, N .
: R - , Q - , I - , Z - , N . , NÌZÌQÌR, IÌR, R=QÈI.
R ( ), .. , , . , .. , , (. . 1.1). " " " ".
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, a £ x £ b, ( ) [; b]; < < b - ]; b[[2]; £<b <£, [; b[ ]; b]. ]-¥; [, ]b; +¥[,
]-¥, +¥[, ]-¥; ] [b; +¥[. b .
( ) - , , -, :
:
1) || ≥ 0 ( );
2) | + y| £ || + |y|;
3) | - y| ≥ || - |y|;
4) |y| = |||y|;
5) |/y| = ||/|y|.
| - | < , > (. . 1.2). , , | - | < e (
e > 0) ] - e, + e[.
. , , .
] - e, + e[, .. , | - | < e ( e > 0) e- (. . 1.3).
() , .
, .
(), X ( Î X) Y ( Î Y) ( y )..
, X y = f(x).
( ), - , f .
X ( ) , Y - .
X , , .. , = f() .
, [5; +¥[, ( 5 ≥ 0).
. :
) ,
y = f(x). . , .
. , . , y, . x3 + y2 = 2. y > 0 y < 0.
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) , , f(x). , -, .
) - (, y), , - y (. 1.3).
1. . y = f(x) , f(-x) = f(x), , f(-x) = -f(x). .
, = 2 , (-)2 = 2; = 3 , (-)3 = -3. = f(x) = 2 +3 , f(-x) = (-)2 +(-)3 = 2 - 3. f(-x) ¹ f(x) f(-x) ¹ - f(x).
(, = 2), (, = 3).
2. . y = f(x) () , c () .
1, 2 Î X 2 > 1. X, f(x2) > f (1) , f(x2) < f (1).
. (.. f(x2) ≥ f (1) f(x2) £ f (1)), .
, = 2 (.. ]-¥; 0]) .
3. . y = f(x) X, , . ( > 0: |f(x)| £ M Î X)
.
, = cos , |cos | £ 1. = ]-¥; +¥[.
, , , .
4. . y = f(x) ¹ 0, f( + ) = f(x).
[3], = sin = 2p,
sin ( +2p) = sin .
. y = f(x) 1 ¹ 2 y1 ¹ y2, x = j(y), y = f(x) . f f-1.(
(-1)).
, .
, = x=lga ( = lg).
(
y = x) (. . 1.3).
. y = f (u) u, U c Y, u, , u = j () , X U. X y = f [j ()] ( , , ).
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, = lg sin , = lg u, u = sin .