¿ô³Ø³Ø½øÏÀ¥Î¡¼¥È

2011

i N Z Q R C A def B, A B.

ii..,.,.. (, ), ( ),.?????????,.

iii 04-13 04-20 04-27 05-04 [ ] 05-11 05-18 05-25 06-01 06-08 06-15 06-22 06-29 07-06 07-13 07-20 07-27 08-03 10-05 10-12 10-19 [ ] 10-26 11-02 11-09 11-16 11-23 [ ] 11-30 12-07 12-14 12-21 12-28 01-11 01-18 2 01-25 13

iv 02-01 02-08,, [8] p.1 p.51. p.1 p.20, p.57 p.166.,. 2..,.,.

v 1................................. 1 2................................. 4 3.............. 5 4..................................... 11 5,,...................... 14 6................................. 19 7............................. 25 8................................... 30 9..................................... 33 10,,............................. 35 11,................................... 41 12............................. 47 13.................................. 58 14,............................... 65 15,................................. 71 16................................. 78 17................................. 88 18................................ 94 101

1 1,,. (, ).,,. 1.. (i). (ii),. (iii),. 2.. 2,, ( [8, p.2] ),. ( [7].),,. (,, ).,,,. 1. (i),,.,,.. (. )

2,,,,,,. (ii),,.,,. (. 1 2,,.) (iii),. ε-δ, [2]. 1.1. 1. {a n }, {b n }, lim a n = α, lim b n = β n n, lim a nb n = αβ, n,,.,.,. 2. y = f(x) [a, b], f(a) f(b). k f(a) f(b) f(c) = k, a < c < b c 1 f(b) y k f(a) a c b x

1 3,,. ( ).

4 2.. 2.1 ([8, p.19]). N : (natural number)., 1, 2, 3,.... 0,. Z : (Zahlen, ).,..., 2, 1, 0, 1, 2,.... Q :.. R : (real number). C : (complex number). ( 0 ) rational number, r r (Quotient) q.. integer. 2.2.. AD [5], N, Z, Q, R.,,, 2.1 (blackboard bold double struck ). [8, p.20],.. 2.3. (definition),., *1. *1 ( ),.

3 5 3,. 3.1. ( ) (proposition). p, q, P, Q. 3.2. 1. 1 + 1 0 ( ) *2. 2. 1 + 1 = 2 ( ) ( ). 3..,. 4. x, x 2 = 2. x (x ). 3.2 4, x,. 3.3., (predicate).,,,.. 3.4. 1., ( ).,,,, *2 1 + 1 = 0 ( ), 1 +,.

6.. for all given any. 2.,.,.,,,,,,,,,, ],, ],.. there exists such that for some,. 3.5. 3.4.2,, 1. 1. 2.,. 3.6.,,. (,, )..,. 1. ( ).

3 7,. 2. ( ).,.. ( 3.7 3.9.) a.,,., (, ), 1. b.,,., (, ), 2.,, (, ),,.,,... P (x) x., x, P (x)., x P (x),,. x, P (x). P (x) x x P (x),,

8.., P (x, y) x, y, x, P (x, y) y., y, x, P (x, y). 3.7., y, ( x, P (x, y) ).,,,. (.) 3.8. x, P (x) x, P (x) x, P (x) x, P (x).. 3.9 ([8, p.12 13]). 1. x, x 2 0., x, x 2 0,. 2. x, x 2 = 2., x 2 = 2 x, x x 2 = 2. x = 2 x 2 = 2,. 3. x, y, y 2 = x. x, ( y, y 2 = x ) x, ( y, y 2 = x ).

3 9 y, y 2 = x, y 2 = x y, y y 2 = x, x, y y 2 = x. x, y x y 2 = x. x, y = x, y, y 2 = x.. 4. y, x, y 2 = x. y, ( x, y 2 = x ) y, ( x, y 2 = x ), ( x, y 2 = x ) y. x, y 2 = x, x y 2 = x, x y 2 = x, y, x y 2 = x, (x y 2 = x ) y. y,., y. 1 y 2 = 1. 2 y 2 = 2. 1 = y 2 = 2, 1 2. 3.10.,.,.,. 3.11. 3.9.3,4,.! 3.12. [8, p.7],,.,,

10,,. 3.13.. 0. 1. x, y, x < y. 2. x, y, x < y. 3. x, y, x < y. 4. x, y, x < y. 3.1. 3.13 1 2, 3 4. 3.2.. 3.13.3 ( 3.11 ). 1. y, x, x < y. 3.3.. 3.13.1 1, 3.13.2 2. 1. x, y, x y. 2. x, y, x y. 3.4.. 1. x, y, z, z(x + y) = 0. 2. x, y, z, z(x+y) = 0. 3.14., x, y, z, i, j, k, l, m, m, n. ([7].)

4 11 4 4.1 ([8, p.28]). 1. (set). A, B, S, T. 2. S, S S (element). x S, x S, x S, S x, x S S x. x S, x S, x S, S x, x S S x. 4.2 ([8, pp.28 30]). A, B. 1. A x, x B, A B (subset), A B B A., A B, A B., A B, B A. 2. A B B A, A B, A = B., A B, A B. 4.3. 4.2.1,. A A (A x, x A ). A B, A B, A B (proper subset).,. A B A B A B A B A B A B 4.4. 4.1.2, 4.2.1,,,,,,.

12 4.5 (. [8, pp.28 29]). 1. (extensional), {}.. {1, 2, 3, 4, 5},,..... {1, 2, 3,..., 10000} (10000 ) 2. (intensional). P (x) x. P (x) x {x P (x)}. x U, P (x) x ( x U) {x U P (x)}.. {n N n 10000} (10000 ) 4.6 ([8, p.31 (3)]). 2.1. N = {1, 2, 3,... } = { n n } Z = {0, ±1, ±2,... } = { n n } Q = { r r } { m } = n m, n Z n 0 4.7.. 1. {1, 2, 3} = {2, 3, 1} = {3, 1, 2} = {1, 3, 2} = {3, 2, 1} = {2, 1, 3} 2. {1, 1, 2} = {1, 2} 3. {1, 2, 3,..., 10000} = {n N n 10000} 4. { x i N, x = ( 1) i } = {1, 1} 5. { x C x 2 = 1 } = {± 1} 4.8. 1., 4.7.2,,,,.,. 2. 4.7.4, { ( 1) i i N }.. 4.7.5 C R { x R x 2 = 1 }. x 2 = 1,.

4 13. 4.9 ( [8, p.31]).. (empty set),., S, S. ( (4.2.1) S, 12.9.,.) 4.10.,,. S 1 = { 1 n n N } S 2 = {x R 0 < x 1} S 3 = {x Q 0 < x 1} S 4 = {x Z 0 < x 1} S 5 = { x R x 2 = 1 } S 6 = { x N x 2 = 1 } S 7 = { x R x 2 = 1 2 S 8 = { x R x 2 = 1 4 S 9 = { x Q x 2 = 1 2 S 10 = { x Q x 2 = 1 4 S 11 = {1} S 12 = S 13 = { ( 1) i i N } x > 0} x > 0} x > 0} x > 0} 4.11.. 1. { 1 n n N n 10 } 2. { x Q 0 < x < 100 x N } 3. { x R x 2 Z x 3 }

14 5,,. 5.1 ([8, p.3]). 1. p, q,, p q. 2. P, Q,,, P Q. 5.2. [8],,. [4]. p,q, p, p q, p q.,.,,,.,,,,.,,,.,,. p, p q, p q, p, q,. 5.3 ([8, pp.3 5])., T, F,. 1. p p p.

5,, 15 p p. p p T F F T ( p (truth table).) 2. p q p q p q. p, q p q. p q p q T T T T F T F T T F F F ( p q (truth table).) 3. p q p, q p q.,, p q p q. p q p q T T T T F F F T F F F F ( p q (truth table).) 5.4.,.,,. 5.5.. p, p p q, p q. ( p p q.),,,

16,,..,, ( ). 5.6. 1. p p p p p T F T F T F, p p. 2. p q q p p q q p q p p q T T T T T T T F F T T T F T T F T T F F F F F F p, q.,. p q, p, q,. q p, q, p,., p, q. 3. p q r p q r q r p q r T T T T T T T F F T T F T F T T F F F T F T T T T F T F F F F F T F F F F F F F 5.3., P (x) x, Q(y)

5,, 17 y, x P (x) x, y P (x) Q(y), P (x) Q(y). P (x) Q(y). P (x) Q(y) x, y (a b ) P (a) Q(b) P (a) Q(b). P (x), Q(y), P (a), Q(b), P (a) Q(b) 5.3.2. P (x) Q(y), x, y., x, y, P (x) Q(y) x, P (x) Q(x). (.) x, y, P (x) Q(y) x, P (x) Q(x),.,, ( ) x, y, P (x) Q(y) x, P (x) Q(x). 5.7. x, x 2 > 0 x 0. x. x 0 x 2 > 0. x = 0 x 0. x 2 > 0 x 0.. (, x 2 0( x 2 0) x = 0 0..) 5.8..

18 1. p p. 2. p p. 3. p q p r. 5.9.. 0. ( 3.13.) 1. 1 + 1 = 2 1 + 1 = 3. 2. 1 + 1 = 2 1 + 2 = 3. 3. x, y, x < y x, y, x < y. 4. x, y, x < y x, y, x < y. 5. x, y, x < y. 6. x, y, x < y. 7. x, y, x < y x > y.

6 19 6 p, q, p q.,, p q.,,.. p q, p q. p q, p q. p p q.,,,, p, q, p q.,. 6.1 ([8, p.5]). p, q, p q (, p q ). p q p q. p q. p q p q T T T T F F F T T F F T ( p q (truth table).) p p q (q ).,. (, [8, p.5 ]..).,. p q, p q, p q, p q p q, T F,

20 p q p q T T T T F F F T F F,,,. p q p q p q p q p q T T T T T T T F F F F F F T T F T F F F T T F F, (p q) q, (p q) (p q). p q q p q., p q,, (p q) (q p). p q p q, p q q p.. p q. p q p q p q, ([8, p.6]).. p q p (, http://www.math.h.kyoto-u.ac.jp/~takasaki/edu/logic/index.html.) 6.2 ([8, p.17]). p q, p, q, q p. q p, p q (contraposition)..

6 21 p q q p p q p, q q p T T F F T T T T F T F F F F F T F T T T T F F T T T T T p, q, 3. ) p q p p q p, q T T F T T T F F F F F T T T T F F T T T, p q p, q. q p (q ), (p )] q (p )] p, q] p q. 5.6.1, 2. 6.3. (theorem), ( ).,,, (proposition), (lemma), (corollary)., 3.1. 3.1. ( 5),,.,,. 6.4. x,., x. 1. x > 2 x > 0. 2. x > 0 x > 2.

22 6.4.1,2,., i. x > 2 x > 0. ii. x > 0 x > 2., i, ii,.. 6.4.1,2 x, x., x x. x a. a, a > 2 a > 0, a > 0 a > 2. a a > 2 a > 0 a > 2 a > 0 a > 0 a > 2 a 0 F F T T 0 < a 2 F T T F a > 2 T T T T 1. x > 2 x > 0. a a > 2 a > 0., x, x > 2 x > 0. x, x > 2 x > 0. (,, x, P (x) Q(x),,.) 2. x > 0 x > 2. a,., x, x > 2 x > 0, x, x > 2 x > 0., i, ii, ( ) i. x, x > 2 x > 0 ii. x, x > 2 x > 0

6 23.,, ( )., ( ),,. 6.5. P (x), Q(x) x P (x) Q(x), ( ), x, P (x) Q(x).,,,,. 6.6. x, P (x) Q(x),, x a P (a) Q(a), a, x. P (a) Q(a), P (a), P (a)., P (a) Q(a). ( [4], [7].) 6.7. x, x 2 > 0 x 0. 5.7,., 6.2. 6.2, x, x 2 0 x 0,. x. x 2 0, x = 0, x 0.. 6.8. ((p q) p) q.

24 6.9.. 0. 1. 1 + 1 = 2 1 + 1 = 3. 2. 1 + 1 = 3 1 + 1 = 4. 3. x, y, x y x < y. 4. x, y, x y x < y.

7 25 7 7.1 ([8, pp.32 33]). S, T. 1. S T S T = { x x S x T }, S T (intersection). 2. S T S T = { x x S x T }, S T (union). 7.2. A, B, A 1, A 2, B 1, B 2.. 1. A B A. 2. A A B. 3. A B 1 A B 2, A B 1 B 2. 4. A 1 B A 2 B, A 1 A 2 B.., 4. A 1 B A 2 B. x A 1 A 2., x A 1 x A 2. (i) x A 1. A 1 B, x B. (ii) x A 2. A 2 B, x B. x B. A 1 A 2 B. 7.3. 1. A B, (i) A B = A. (ii) A B = B. 2. A i B i (i = 1, 2), (i) A 1 A 2 B 1 B 2. (ii) A 1 A 2 B 1 B 2.., 7.2.

26 1. (i) 7.2.1, A B A., A A, A B, 7.2.3, A A B., ( 4.2.2 ) A B = A. 2. (i) 7.2.1, A 1 A 2 A 1, A 1 B 1, A 1 A 2 B 1. A 1 A 2 B 2., 7.2.3, A 1 A 2 B 1 B 2.,. 7.4., (Venn diagram),,.. A = {1, 2}, B = {{1, 2}, 1}, C = {1, 2, 3}.?,,., A = Q, B = { r + s 2 r, s Q }, C = { r + s 3 r, s Q }. 7.5.. 1. ( ) 2. ( ) A B = B A A B = B A 3. ( ) A (B C) = (A B) C A (B C) = (A B) C A (B C) = (A B) (A C) A (B C) = (A B) (A C)

7 27. 1, 2,,. 2, { x x A x B x C }. ( (p q) r p (q r).). A (B C) = (A B) (A C) (a) A (B C) (A B) (A C). x A (B C)., x A x B x C. (i) x B. x A x B, x A B (A B) (A C). (ii) x C. x A x C, x A C (A B) (A C). x (A B) (A C). A (B C) (A B) (A C). (b) (A B) (A C) A (B C). B B C, A B A (B C)., C B C, A C A (B C). (A B) (A C) A (B C). A (B C) = (A B) (A C), (A B) (A C) = ( (A B) A ) ( (A B) C ) = A ( (A B) C ) = A ( (A C) (B C) ) = ( A (A C) ) (B C) = A (B C).,,. (a) A (B C) (A B) (A C). B C B, A (B C) A B., B C C, A (B C) A C., A (B C) (A B) (A C). (b) (A B) (A C) A (B C). x (A B) (A C)., x A B x A C.

28 (i) x A. x A A (B C). (ii) x A. x A B, x A x B, x A, x B., x A C, x A x C, x A, x C., x B x C, x B C A (B C)., x A (B C). (A B) (A C) A (B C)., 5. 5.6.3, 5.8, p (q r) (p q) (p r)., A (B C) = { x x A x B C } = { x x A (x B x C) } = { x (x A x B) (x A x C) } = { x x A B x A C } = (A B) (A B). 7.6. (b). 1. x A, x A.. x A (B C)),, x A x B C., x A. x A. x A x B C ( ) x A x B C. 2. x A B, x A, x B,. x A, x B, x A C x C.

7 29, x A x C. x A x B { x A x C 7.7.,., p (q r) (p q) (p r). 7.8. 1. 7.2.3. 2. 7.3.2.(i), 7.2. 3. 7.3.2.(ii), 7.2. 4. 7.5,.

30 8 8.1 ([8, p.34]). 1., X, X (universal set). 2. S X, S c S c = {x X x S}, S (complementary set). 8.2 ([8, p.35]). X, S, T X.. 1. S S c =, S S c = X. 2. (S c ) c = S. 3. c = X, X c =. 4. S T S c T c. S = T S c = T c. 5. S T = S T c S c T. 6. (de Morgan ( ) ) (S T ) c = S c T c (S T ) c = S c T c 8.3.,.,,,.. 1 4. 5. (i) S T =. S = S X = S (T T c ) = (S T ) (S T c ) = (S T c ) = S T c T c. (ii) S T c. S T T c T = S T =. 6. (S T ) c = S c T c

8 31 (i) (S T ) c S c T c. ((S T ) c S) T = (S T ) c (S T ) = (S T ) c S T c. (, (S T ) c = (S T ) c X = (S T ) c (S c S) = ((S T ) c S c ) ((S T ) c S) S c T c. x (S T ) c, x S T. x S c T c,, x S c x T c. x S c *3. x S c. x S *4. x T, x S T. x T, x T c *5.) (ii) S c T c (S T ) c. S S T S c (S T ) c., T S T T c (S T ) c. S c T c (S T ) c. ( ),, (p q) (p ) (q ). (S T ) c = {x X x S T } = { x X x S T } (S T ) c = S c T c = { x X (x S x T ) } = { x X (x S ) (x T ) } = { x X x S x T } = { x X x S c x T c} = S c T c. *3 (S T ) c S c S c. *4 x (S T ) c S. *5 (S T ) c S T c.

32, (S c T c ) c = (S c ) c (T c ) c = S T. (S T ) c = S c T c. 8.4 ([8, p.34]). A, B. A B A B = { x x A x B } A B (set difference). A \ B. (A B.) 8.5. X, S, T X. S T = S T c., S c = X S... 8.6. X, A, B, C X.. 1. A B C A B c C. 2. A B = X A c B A B c.

9 33 9 9.1 ([8, p.36]). 1. X, Y. X x X Y X Y (map). f X Y, f : X Y. X f Y. 2. f : X Y. f, x X y Y, f x y, f : x y., f x y, y f(x). 9.2. 1.. 2.. Y = R, C, ( ). 3. f(x), f (x). f(x) f x Y. f(x): X Y.. Y. Y ( )., ( ). (X, X.) 9.3. S = {0, 1, 2}, T = {0, 1}. 1. f : S T 0 0 1 1 2 1 2. g : T S 0 0 1 1 3. h: T T 0 0 1 1

34 9.4 ([8, p.37]). X, S X. 1. f : X X (identity map), x X, f(x) = x. X id X id. 2. f : S X (inclusion map), s S, f(s) = s ( s X ). (, S = X.) 9.5. 9.3 g : T S, h: T T T. 9.6 ([8, p.37]). X, Y, Z, f : X Y, g : Y Z., x X, g(f(x)) Z f g (composite map), g f. (g f)(x) = g(f(x)). g f : X x Z g(f(x)) g f : X f Y g Z. 9.7. S = {0, 1, 2}, T = {0, 1}, f, g, h, 9.3. 1. g f, f g? 2. S T. 9.8 (2011/6/28 ). f, g : X Y. x X, f(x) = g(x), f g, f = g. 9.1. f : X Y. f id X = f, id Y f = f.

10,, 35 10,, 10.1 ([8, p.36]). f : X Y (injection, injective map). x 1, x 2 X, x 1 x 2 f(x 1 ) f(x 2 ). def, f : X Y, x 1, x 2 X, f(x 1 ) = f(x 2 ) x 1 = x 2.. 10.2. A def B, A B. 10.3.. x 1, x 2 X, x 1 = x 2 f(x 1 ) = f(x 2 )., f,. 10.4. 1.. 2.. 10.5. 1. f(x) = x 2 f : R R. f(1) = f( 1). 2. g(x) = x 2 g : R 0 R. R 0 := {x R x 0}. x 1, x 2 R 0. g(x 1 ) = g(x 2 ), 0 = g(x 1 ) g(x 2 ) = x 2 1 x 2 2 = (x 1 x 2 )(x 1 + x 2 ) x 1 x 2 = 0 x 1 + x 2 = 0., x 1, x 2 0, x 1 + x 2 = 0 x 1 = x 2 = 0. ( ) x 1 = x 2. 3. h: R R x x 3. x 1, x 2 R. x 1 x 2, x 1 < x 2 x 1 > x 2. h, h(x 1 ) < h(x 2 ) h(x 1 ) > h(x 2 ), h(x 1 ) h(x 2 ).

36 10.6. x 1, x 2 R. h(x 1 ) = h(x 2 ) 0 = h(x 1 ) h(x 2 ) = x 3 1 x 3 2 = (x 1 x 2 )(x 2 1 + x 1 x 2 + x 2 2) x 1 = x 2 *6. 1. 9.3 f. f(1) = f(2). 2. 9.3 g. g(0) g(1). 3. 9.3 h. h(0) h(1). 10.7 ([8, p.36]). f : X Y (surjection, surjective map). y Y, x X, y = f(x). def 10.8. 1.. 2.. 10.9. 1. f(x) = x 2 f : R R. x R, f(x) = x 2 0. 1 < 0, f(x) = 1 x R. 2. f(x) = x 2 f : R R 0. R 0 := {x R x 0}. y R 0, x = y, x R, f(x) = x 2 = y. 10.10. S = {0, 1, 2, 3}. f,. f : S S 0 0 1 0 2 2 3 3. f(0) = f(1).. f(x) = 1 x S. 10.11 ([8, p.39] ). X, Y, Z, f : X Y, g : Y Z.. *6 x 2 1 + x 1x 2 + x 2 2 = 0 x 1 = x 2 = 0.

10,, 37 1. g f f. 2. g f g.. 1. x 1, x 2 X. f(x 1 ) = f(x 2 ) x 1 = x 2. f(x 1 ) = f(x 2 ). (g f)(x 1 ) = g (f(x 1 )) = g (f(x 2 )) = (g f)(x 2 )., g f x 1 = x 2. 2. z Z. y Y z = g(y)., g f, x X, (g f)(x) = z. y = f(x) Y, g(y) = g (f(x)) = (g f)(x) = z. 10.12. f : X Y (bijection, bijective map). def f. 10.13 ([8, p.37]). f : X Y.. 1. y Y, y = f(x) x X. 2. y Y, y = f(x) x X Y X. f 1. x = f 1 (y) f(x) = y. f 1 : Y X. f 1 f = id X, f f 1 = id Y. 10.14. f : X Y., 10.13.2 f 1 : Y X f (inverse map). 10.13. 1. y Y. f, y = f(x) x X. x X f(x ) = y, f(x ) = y = f(x), f, x = x. x.

38 2. (i) f 1 f = id X. x X. ( f 1 f ) (x) = x. y = f(x), f 1, f 1 (y) = x. x = f 1 (y) = f 1 (f(x)) = (f 1 f)(x). (,., f 1 (f(x)), f f(x) ( ). x f f(x) f 1 (f(x)) = x.) (ii) f f 1 = id Y. y Y. ( f f 1) (y) = y, f ( f 1 (y) ) = y,. (f y ( ) f 1 (y).), x = f 1 (y), f(x) = y. y = f(x) = f ( f 1 (y) ) = (f f 1 )(y). 10.15. f : X Y.. 1. f. 2. g : Y X, g f = id X, f g = id Y., g = f 1.. 1 2. 10.13, g = f 1. 2 1. g f = id X, id X, 10.11, f., f g = id Y, id Y, 10.11, f. g : Y X g f = id X. g = g id Y = g (f f 1 ) = (g f) f 1 = id X f 1 = f 1. f 1 2 1), 10.16. f f 1. 10.17. f : N Z, g : Z N,

10,, 39 n 2, n f(n) = n + 1, n { 2 2l, l > 0 g(l) = 2l + 1, l 0 (f(n) Z, g(l) N ), g f = id N, f g = id Z, f, g, g = f 1. 10.18. 1.. 2.. 3.. ( 10.15..) 10.19. A = {1, 2, 3}, B = {1, 2}. 1. A B? 2. A B? 3. A B? 4. A B? 5. B A? 6. B A? 7. B A? 10.20. X, Y, Z, f : X Y, g : Y Z..,. 1. g f g. 2. g f f. 10.21. f : X Y, S., g 1, g 2 : S X f g 1 = f g 2, g 1 = g 2. 10.22. f : X Y, T., h 1, h 2 : Y T h 1 f = h 2 f, h 1 = h 2. 10.23. f : X Y, g, h: Y X. g f = id X, f h = id Y, f, g = h = f 1. 10.24. X, Y, Z, f : X Y, g : Y Z..

40 1. f, g g f. 2. f, g g f. 3. f, g g f. 10.25. m, n N, A = {1, 2,..., m}, B = {1, 2,..., n}. 1. A B? 2. m n, A B? 3. n = 2 m n, A B? 10.21, 10.22,.. 10.1. f : X Y. S, g 1, g 2 : S X, f g 1 = f g 2 g 1 = g 2, f. 10.2. f : X Y. T, h 1, h 2 : Y T, h 1 f = h 2 f h 1 = h 2, f.

11, 41 11, 11.1 ([8, p.36]). f : X Y, A X., Y f(a) f(a) = {f(a) a A} Y, f A (image). 11.2. f : X Y, B Y., X f 1 (B) f 1 (B) = {x X f(x) B} X, f B (inverse image)., f 1 (B), f B X. 11.3. f( ) =, f 1 ( ) =. 11.4. 10.14, 11.2. f : X Y. 1. f, f f 1 : Y X. 2., f, Y B X f 1 (B). 3., f 1,., f 1 ( )., f 1. 11.5. f : X Y, B Y., f 1 (B) 1. f B f 1 (B) 2. f 1 : Y X B (f 1 )(B),, f 1 (B) = (f 1 )(B). 11.6 ([8, p.41]). f : X Y, A 1, A 2 X.,. 1. A 1 A 2 f(a 1 ) f(a 2 ).

42. 2. f(a 1 A 2 ) = f(a 1 ) f(a 2 ). 3. f(a 1 A 2 ) f(a 1 ) f(a 2 ). 4.,. 1. A 1 A 2. y f(a 1 )., a A 1, y = f(a). A 1 A 2, a A 2., y = f(a) f(a 2 ). 2. (i) f(a 1 A 2 ) f(a 1 ) f(a 2 ). y f(a 1 A 2 )., a A 1 A 2, y = f(a). a A 1, y = f(a) f(a 1 ) f(a 1 ) f(a 2 )., a A 2 y f(a 2 ) f(a 1 ) f(a 2 ). y f(a 1 ) f(a 2 ). (ii) f(a 1 ) f(a 2 ) f(a 1 A 2 ). A 1 A 1 A 2, f(a 1 ) f(a 1 A 2 )., A 2 A 1 A 2,f(A 1 ) f(a 1 A 2 ). f(a 1 ) f(a 2 ) f(a 1 A 2 ). 3. A 1 A 2 A 1, f(a 1 A 2 ) f(a 1 )., A 1 A 2 A 2, f(a 1 A 2 ) f(a 2 )., f(a 1 A 2 ) f(a 1 ) f(a 2 ). 4. f : R R, f(x) = x 2, R A 1 = (, 0], A 2 = [0, ). A 1 A 2 = {0} f(a 1 A 2 ) = f({0}) = {f(x) x {0}} = {f(0)} = {0}. (f({0}) f(0), {0} 0.) f(a 2 ) = [0, )., f(a 1 ) f(a 2 ) = [0, ). f(a 1 ) = {f(x) x A 1 } = { x 2 x 0 } = [0, ) f(a 1 A 2 ) f(a 1 ) f(a 2 ).. f(a 1 A 2 ) f(a 1 ) f(a 2 ).

11, 43 f : {0, 1} {0} 0 0 1 0. {0}, {1} {0, 1}, {0} {1} =, f({0} {1}) =., f({0}) = f({1}) = {0}, f({0}) f({1}) = {0}. 11.7 ([8, p.37]). f : X Y f(x) = Y.. f. f(x) Y. Y f(x). y Y. f, x X, y = f(x)., y = f(x) f(x). f(x) = Y. y Y. y Y = f(x) = {f(x) x X}, x X, y = f(x). f. 11.8 ([8, p.41]). f : X Y, B 1, B 2 Y.,. 1. B 1 B 2 f 1 (B 1 ) f 1 (B 2 ). 2. f 1 (B 1 B 2 ) = f 1 (B 1 ) f 1 (B 2 ). 3. f 1 (B 1 B 2 ) = f 1 (B 1 ) f 1 (B 2 ). ( 11.6.3), 3.. 1. x f 1 (B 1 ). f(x) B 1. B 1 B 2, f(x) B 2., x f 1 (B 2 ). 2. (i) f 1 (B 1 B 2 ) f 1 (B 1 ) f 1 (B 2 ). x f 1 (B 1 B 2 ). f(x) B 1 B 2. f(x) B 1, x f 1 (B 1 ) f 1 (B 1 ) f 1 (B 2 ). f(x) B 2, x f 1 (B 2 ) f 1 (B 1 ) f 1 (B 2 ). x f 1 (B 1 ) f 1 (B 2 ). (ii) f 1 (B 1 ) f 1 (B 2 ) f 1 (B 1 B 2 ). B 1 B 1 B 2, f 1 (B 1 ) f 1 (B 1 B 2 )., B 2 B 1 B 2, f 1 (B 2 ) f 1 (B 1 B 2 )., f 1 (B 1 ) f 1 (B 2 ) f 1 (B 1

44 B 2 ).. f 1 (B 1 B 2 ) = {x X f(x) B 1 B 2 } = { x X f(x) B 1 f(x) B 2 } = { x X x f 1 (B 1 ) x f 1 (B 2 ) } = f 1 (B 1 ) f 1 (B 2 ). 3. (i) f 1 (B 1 B 2 ) f 1 (B 1 ) f 1 (B 2 ). B 1 B 2 B 1, f 1 (B 1 B 2 ) f 1 (B 1 )., B 1 B 2 B 2, f 1 (B 1 B 2 ) f 1 (B 2 ). f 1 (B 2 ). (ii) f 1 (B 1 ) f 1 (B 2 ) f 1 (B 1 B 2 )., f 1 (B 1 B 2 ) f 1 (B 1 ) x f 1 (B 1 ) f 1 (B 2 ). x f 1 (B 1 ) x f 1 (B 2 ). x f 1 (B 1 ) f(x) B 1. x f 1 (B 2 ) f(x) B 2., f(x) B 1 B 2., x f 1 (B 1 B 2 ).. f 1 (B 1 B 2 ) = {x X f(x) B 1 B 2 } = { x X f(x) B 1 f(x) B 2 } = { x X x f 1 (B 1 ) x f 1 (B 2 ) } = f 1 (B 1 ) f 1 (B 2 ). 11.9. f : X Y, A X, B Y.,. 1. f 1 (f(a)) A. 2. f f 1 (f(a)) = A. 3. f(f 1 (B)) B. 4. f f(f 1 (B)) = B.. 1.. (A f f(a).), a A. f(a) f(a)., a f 1 (f(a)).. f : R R, f(x) = x 2, f 1 (f({1})).

11, 45 2. f., f 1 (f(a)) A. x f 1 (f(a)). f(x) f(a), a A, f(x) = f(a). f, x = a A. 3.. (f 1 (B) f B.), y f(f 1 (B))., x f 1 (B), y = f(x). x f 1 (B) f(x) B., y = f(x) B.. f : R R, f(x) = x 2, f(f 1 (R)). 4. f., B f(f 1 (B)). b B. f, x X, f(x) = b. f(x) = b B, x f 1 (B)., b = f(x) f(f 1 (B)). 11.10. f : X Y, B Y., f B f 1 (B), f 1 : Y X B (f 1 )(B), f 1 (B) = (f 1 )(B). 11.11. f : X Y, A 1, A 2 X. f, f(a 1 A 2 ) = f(a 1 ) f(a 2 ). 11.12. f : X Y, B Y., B f(x) = f(f 1 (B)). 11.13. f : X Y, A X. 1. f(x A) f(x) f(a). 2. f,,, f(x A) = f(x) f(a). 3. f(x A) f(x) f(a). 11.14. f : X Y, B Y., f 1 (Y B) = X f 1 (B).

47 12 12.1. p, q. 1. p p. 2. p q p q. 3. p q p q.,,, (logical connective). 12.2. [8] p, p.,.. (negation). (disjunction). (conjunction). (implication). 12.3 ([8, p.7]). P (x) x. 1. x, P (x). All A.,. x : P (x) 2. x, P (x). x : P (x)

48 Exist E.,., (quantifier).. (universal quantifier),. (existential quantifier),.,,,. P (x), x, P (x).,.,.,,,. 12.4.,. P (x, y) x, y, x, P (x, y) y. 12.3,. x : P (x, y) y, x, P (x, y) 12.3, y : x : P (x, y) y : ( x : P (x, y)), :,. y, x : P (x, y)

12 49 12.5. (,,, ) (, ) (logical symbol),. 12.6..,,,..,,,,,,.,,,,. x x, x, x 3 0, x, x 2 = 2, x x. 12.7. P (x), Q(x) x. 1. P (x) x, Q(x) x(p (x)) : Q(x)., x. 2. P (x) x, Q(x) x(p (x)) : Q(x)., x..,.,.,,., x : P (x) Q(x) x(p (x) Q(x)) ( ).,. 12.8. P (x), Q(x) x..

50 1. x(p (x)) : Q(x) x : P (x) Q(x). 2. x(p (x)) : Q(x) x : P (x) Q(x).,.,,,,,. 12.9. P (x) x.. 1. x : P (x). 2. x : P (x).. 1. x : x P (x). x, x. x, x P (x)., x, x P (x). 2. x : (x ) P (x). x x. (x ) P (x) x., x, (x ) P (x). 12.10 ([8, p.12 13], 3.9). 1. x, x 2 0. x(x R) : x 2 0., x R : x 2 0. 12.8 x : x R x 2 0. x, x x 2 0, x x 2 0 6.5. 6.5,.

12 51. 2. x, x 2 = 2. x R : x 2 = 2 x : (x R) (x 2 = 2).. 3. x, y, y 2 = x. x(x R x > 0), y R : y 2 = x x R(x > 0), y R : y 2 = x.,,,,, [8, p.13, ]., x > 0, y R : y 2 = x.. 4. y, x, y 2 = x. y R, x > 0 : y 2 = x.. 12.11. 1. 3.11, x > 0, y R : y 2 = x y R, x > 0 : y 2 = x. (.)!

52 2.,. ( ). 3. 4.5). (i) 4.8,.,,. f : X Y B Y f 1 (B) = { x X f(x) B}. (ii) {x P (x)}, P (x) x.,,. f : X Y A X f(a) = {f(x) x A}., x A,. x x. (iii),.,. {x R ε > 0 : x ε} {x R ε > 0 : x ε} = {x R x 0} 12.12. 12.10.3,4, 12.8,,. 3. x > 0, y R : y 2 = x = x(x > 0) : y(y R) : y 2 = x x(x > 0) : y : (y R) (y 2 = x) x : (x > 0) ( y : (y R) (y 2 = x)) x, y 2 = x y. y 12.14. x : y : (x > 0) ((y R) (y 2 = x)) = x, y : (x > 0) ((y R) (y 2 = x)).

12 53 12.8. 12.10.1,,,,,.,, 6.5,. 12.13.,,., 12.14,12.15 12.18,12.16.., p x, x, x p, p, p (1 6), x, x p, p (7 10). 12.14. p (, x ), Q(x) x.. 1. x : p Q(x) p ( x : Q(x) ). 2. x : p Q(x) p ( x : Q(x) ). 3. x : p Q(x) p ( x : Q(x) ). 4. x : p Q(x) p ( x : Q(x) ). 5. x : p Q(x) p ( x : Q(x) ). 6. x : p Q(x) p ( x : Q(x) ). 7. x : ( Q(x) p ) ( x : Q(x) ) p. 8. x : ( Q(x) p ) ( x : Q(x) ) p. 9. x : ( Q(x) p ) ( x : Q(x) ) p. 10. x : ( Q(x) p ) ( x : Q(x) ) p.. 3. q, p p q, p p q q. p., p ( x : Q(x) )., x, p Q(x), x : p Q(x). p. x : Q(x).

54, p ( x : Q(x) )., x : Q(x), x, Q(x)., x, p Q(x)., x : p Q(x). x : Q(x)., p ( x : Q(x) )., x : Q(x), Q(x 0 ) x 0. x 0, p Q(x 0 )., x : p Q(x). q, p p q q, p p q, 4. 6, 7, p q q p,,. 1, 2, p q p q,,. 5, 6, 9,10 3, 4., 13.1, 13.4, 3, 4. 5. ( ) x : p Q(x) x : ( p Q(x) ) 13.4 x : p Q(x) 13.1 p ( x : Q(x) ) 4 p ( x : Q(x) ) 13.4 ( p ( x : Q(x) )) 13.1. 13.1 ( ) x : p Q(x) x : p Q(x) (p ( x : Q(x) )) p ( x : Q(x) ). 12.15. P (x), Q(x) x.. 1. x : P (x) Q(x) ( x : P (x) ) ( x : Q(x) ). 2. x : P (x) Q(x) ( x : P (x) ) ( x : Q(x) )...

12 55 x : P (x) Q(x)., x P (x) Q(x). P (x) Q(x), P (x), Q(x).,, x P (x). x : P (x). x : Q(x)., ( x : P (x)) ( x : Q(x)). ( x : P (x)) ( x : Q(x))., x : P (x), x : Q(x). x : P (x), x P (x). x Q(x)., x, P (x), Q(x), P (x) Q(x)., x : P (x) Q(x). 2, 13.1, 13.4 1. x. 12.16. P (x), Q(x), R(x) x.. 1. x(r(x)) : P (x) Q(x) ( x(r(x)) : P (x) ) ( x(r(x)) : Q(x) ). 2. x(r(x)) : P (x) Q(x) ( x(r(x)) : P (x) ) ( x(r(x)) : Q(x) ).. p, q, r, r (p q) (r p) (r q) r (p q) (r p) (r q). 12.17. 12.15 12.16.,. 1. (i) x R : (x > 0) (x 0). (ii) ( x R : x > 0) ( x R : x 0). 2. (i) x R : (x > 0) (x 0). (ii) ( x R : x > 0) ( x R : x 0). 12.14 x,. ( 2, 4, 5, 8, 9 ) 12.18. p (, x ), Q(x), R(x) x.. 1. x(r(x)) : p Q(x) p ( x(r(x)) : Q(x) ).

56 2. x(r(x)) : p Q(x) ( p ( x(r(x)) : Q(x) )) ( x : R(x)). 3. x(r(x)) : p Q(x) p ( x(r(x)) : Q(x) ) (. 4. x(r(x)) : p Q(x) p ( x(r(x)) : Q(x) )) ( x : R(x)). ( 5. x(r(x)) : p Q(x) p ( x(r(x)) : Q(x) )) ( x : R(x)). 6. x(r(x)) : p Q(x) p ( x(r(x)) : Q(x) ). 7. x(r(x)) : ( Q(x) p ) ( x(r(x)) : Q(x) ) p. 8. x(r(x)) : ( Q(x) p ) ( ( x(r(x)) ) ) : Q(x) p ( x : R(x)). 9. x(r(x)) : ( Q(x) p ) ( ( x(r(x)) ) ) : Q(x) p ( x : R(x)). 10. x(r(x)) : ( Q(x) p ) ( x(r(x)) : Q(x) ) p.. 12.14, 12.8, 12.14, 12.15. 7,8. 7. r (q p) r (q p) ( r q) p (r q) p (1). x(r(x)) : ( Q(x) p ) ( x : R(x) ( Q(x) p )) 12.8 ( (R(x) ) ) x : Q(x) p (1) ( x : ( R(x) Q(x) )) p 12.14 ( x(r(x)) : Q(x) ) p 12.8. 8. (r q) r r (2).

12 57 x(r(x)) : ( Q(x) p ) x : R(x) ( Q(x) p ) 12.8 x : ( R(x) Q(x) ) ( R(x) p ) ( ) ( ) x : R(x) Q(x) x : R(x) p 12.15 ( ) ( ( x ) ) x : R(x) Q(x) : R(x) p 12.14 ( ( x ) ( ) ) ( ( x ) ) : R(x) Q(x) x : R(x) : R(x) Q(x) p ( x : ( R(x) Q(x) ) ) ( ( x ) ) R(x) : R(x) Q(x) p 12.15 ( ) ( ( x ) ) x : R(x) : R(x) Q(x) p (2) ( ) ( ( x(r(x)) ) ) x : R(x) : Q(x) p 12.8 ( ( x(r(x)) ) ) ( ) : Q(x) p x : R(x)., 7, 8. 12.19. 12.14 7 7, 12.14., (r (q p) (r q) p), (r (q p) (r q) (r p)). 12.20. E = { x Z x }, O = { x Z x }. 1.. (i) x E, y O : x y. (ii) x E, y O : x y. (iii) ε > 0, N N, n N : (n N) ( a n a ε). 2.. (i) x, y, x < y. (ii) x, y, x < y. (iii) ε, N, n N n N, a n a < ε. (iv) ε, N, n N a n a < ε.

58 13 13.1. [8, pp.14 16] p, q.. 1. ( p) p. 2. (de Morgan ( ) ) (i) (p q) ( p) ( q). (ii) (p q) ( p) ( q). 3. (p q) p ( q)... 1 5.6.1. 2, 3.), 2, 1, ( 8.2 ). 3, 6.2 1,2. (p q) (( p) q) ( p) ( q) p ( q) 13.2.,,, 3. 3 p q, p q,.. 13.3. p q p q (.) 13.4. [8, p.14] P (x) x.. 1. ( x : P (x) ) x : P (x). 2. ( x : P (x) ) x : P (x).... 13.5. [8, p.16] P (x), Q(x) x..

13 59 1. ( x(p (x)) : Q(x) ) x(p (x)) : Q(x). 2. ( x(p (x)) : Q(x) ) x(p (x)) : Q(x)..,, ( x(p (x)) : Q(x) ) ( x : P (x) Q(x) ) x : ( P (x) Q(x) ) x : P (x) Q(x) x(p (x)) : Q(x). 2,,. 13.1,13.4,13.5. 13.6. [8, p.15] Step 1 (:),. Step 2, Step 3.,..) Step 3 ( ) 13.1. ( Step 1.) 13.7. x > 0, y R : y 2 = x. ( x > 0, y R : y 2 = x ) x > 0, y R : y 2 x 12.12, x > 0, y R : y 2 = x,, x : (x > 0) ( y : (y R) (y 2 = x)) *7. ( x : (x > 0) ( y : (y R) (y 2 = x)) ) x : ( (x > 0) ( y : (y R) (y 2 = x)) ) x : x > 0 ( y : (y R) (y 2 = x)), *7,.

60 ( y : (y R) (y 2 = x)) y : ((y R) (y 2 = x)) y : (y R) (y 2 = x) y : (y R) (y 2 x) ( x : (x > 0) ( y : (y R) (y 2 = x)) ) x : x > 0 ( y : (y R) (y 2 = x)) x : x > 0 y : (y R) (y 2 x).,,,. x > 0 : y : (y R) (y 2 x) x > 0 : y : (y R) (y 2 x) x > 0 : y R : y 2 x x > 0, y R : y 2 x,,. 13.8.. Step 1 Step 2 Step 3. 13.6. Step 2. 13.9., ( ) Step 1..,,,.,,,,. 13.10.. 1. A B

13 61 4.2, A x, x B. Step 1 x A : x B. Step 2 x A : x B. Step 3, x A, x B.,, A, B x. ( x A : x B x : x A x B.) ( A B, A B, A B,...) 2. α x 4 = 1, x 3 = 1. ( α).,,, α ( 6.5). Step 1, α R : α 4 = 1 α 3 = 1. Step 2, α R : α 4 = 1 α 3 1. Step 3, α, α 4 = 1 α 3 1.,, α, α x 4 = 1, x 3 = 1. x 4 = 1, x 3 = 1 α.,.., α 4 = 1 α, α 3 = 1, Step 1, α R(α 4 = 1) : α 3 = 1. Step 2, α R(α 4 = 1) : α 3 1. Step 3, α 4 = 1 α, α 3 1. x 4 = 1, x 3 = 1 α..

62 13.11. a *8., α. 1. x > a α x, α a. 2. x < a α x, α a... x > a α x, α a. x < a α x, α a... 1. α R : ( x > a : α x) α a. ((p q) ( q p)), α R : α a ( x > a : α x). x > a : α x x > a : α x, α R : α > a ( x > a : α > x). α > a. x = (α + a)/2, x > a, α > x. 2. 1.. α R : α > a ( x > a : α > x) α > a, x > a x, α > x. *i α > a, α > x > a x. 1,. *8, a,.

13 63 α R : α > a ( x > a : α > x) α > a : ( x > a : α > x) α > a, x > a : α > x. *i α R : α > a ( x > a : α > x) α R : α > a ( x : (x > a) (α > x)) α R : α > a ( x : α > x > a). 13.12. α. ε > 0 α ε, α 0.. 13.11 a = 0. 13.13. α. ε > 0 α < ε, α 0.. α < ε α ε. 13.14. {x R ε > 0 : x ε} = {x R x 0}.. (x 0 0 < ε, x ε.) 13.12. 13.15 ([8, p.50]). α. 1. ε > 0 α < ε, α = 0. 2. ε > 0 α ε, α = 0.. α 0, α = 0 α = 0. 13.16. p, q.. 1. (p q) ( p) ( q).

64 2. (p q) ( p) ( q). 3. (p q) p ( q). 13.17. p, q. p q, (p q). 13.18.. 1. f : X Y. 2. f : X Y. 3. ε, N, n N n N, a n a < ε. 4. ε, N, n N a n a < ε. 5. ε, δ, x R, 0 < x x 0 < δ f(x) a < ε.

14, 65 14,,,., [3],[9],,... R ( ). 1. ( ) a R : a a 2. ( ) a, b R : a b b a a = b 3. ( ) a, b, c R : a b b c a c (order), (ordered set). R. 4. a, b R a b b a. 4 (total order) (linear order). R. a b a b, a < b. a b b a. 14.1. [8, p.10], a < b a = b a b,, <,,.,, a b a < b a = b. a b a b (a b a = b) (a b a b) (a b a = b) a < b a = b, R.

66 14.2. a, b (a < b) [a, b] := {x R a x b} a, b (closed interval). (a, b) := {x R a < x < b} a, b (open interval). (, a] := {x R x a} (, a) := {x R x < a} [a, ) := {x R x a} (a, ) := {x R x > a}. 14.3 ([8, p.57]). A R. 1. m R A (upper bound) x A, x m. def m A. x A : x m ( x : x A x m). 2. l R A (lower bound) x A, l x. def l A. x A : l x. 3. A A (bounded from above). m R, x A : x m. 4. A A (bounded from below). l R, x A : l x.

14, 67 5. (bounded). A l R, m R, x A : l x m. 14.4.,. 14.5. a, b R, a < b. (a, b) U((a, b)), L((a, b)). U((a, b)) := { x R x (a, b) } L((a, b)) := { x R x (a, b) }., U((a, b)) = [b, ), L((a, b)) = (, a].. L((a, b)) (, a]., l (, a], l a. x (a, b), a < x, l x. l (a, b),, l L((a, b)). L((a, b)) (, a]. l L((a, b)), l (a, b)., x > a l x., x < b, x (a, b),, l x. x b, c = (a + b)/2, c (a, b), l c. c < b, l c < b x l x., 13.11, l a, l (, a]. L((a, b)) = (, a]. U((a, b)) = [b, ). 14.6 ([8, p.44]). A R. 1. M R A (maximum number) (i) M A def (ii) M A. M A (M A) ( x A : x M). (M A) ( x : x A x M). M = max x = max x = max A. x A A

68 2. m R A (minimum number) (i) m A def (ii) m A. m A (m A) ( x A : m x). m = min x = min x = min A. x A A 14.7. a, b, a < b. max[a, b] = b, min[a, b] = a. max(a, b), min(a, b).. max[a, b] = b, min[a, b] = a. min(a, b). m (a, b), x (a, b), x < m. *i m (a, b)., a < m < b. x = (a + m)/2, a < x < m., x < m., m < b, a < x < b,, x (a, b).,,.. *i min(a, b),, m (a, b), x (a, b) : x < m. m R : m (a, b) ( x (a, b) : m x) m (a, b) : ( x (a, b) : m x) m (a, b), x (a, b) : m x.. m (a, b),, m (a, b), m (a, b)., (a, b), m., m (a, b), m (a, b)., m (a, b), m (a, b)... min(a, b), 14.5. L((a, b)) (a, b). min(a, b), m L((a, b)), m (a, b) *i, 14.5,

14, 69 L((a, b)) = (, a],. *i min(a, b), m R : m (a, b) m L((a, b)) m R : m L((a, b)) m (a, b) m L((a, b)) : m (a, b)., m L((a, b)) : m (a, b). 14.8 ([8, pp.49 50]). A R ( ).., M 1, M 2 A. (i1) M 1 A (ii1) a A : a M 1 (i2) M 2 A (ii2) a A : a M 2 (i1) (ii2) M 1 M 2. M 2 M 1. M 1 = M 2.. 14.9. A R, l, m, M R. 1. m A. 2. l A. 3. M A. 4. A. (.) 5. A. (.) 6. 5. 14.10. a, b R, a < b. 1. max[a, b] = b, min[a, b] = a. 2. max(a, b). 14.11. A R., max A, min A, A.

70 14.12. A R. A M R, x A : x M.

15, 71 15,, R (a, b)., b, (a, b), a,.,,. 15.1 ([8, p.57]). A R. 1. A, A (supremum) sup a sup A a A. A, sup A = min U(A). U(A) := { x R x A } 2. A, A (infimum). A, inf A = max L(A). inf a inf A a A L(A) := { x X x A },,,,.. 15.2 (, [8, p.58]). R. 1.,. 2.,. 15.3.,,.,,,,

72.,,.,.,,, N, N Z, Z Q, Q R, R,. [6], [1].. 15.4. 1.,,., m R A R, a A : a m. A =, 12.9, m R, a : a m., m R, m.,, U( ) = R. L( ) = R. R, sup, inf. 2. A R, sup A.,, U(A) =.. 3. A R, inf A. 15.5 ([8, pp.61 62]).,..,.,. 14.8,. 15.6 ([8, p.62]). A R. 1. max A A,, sup A = max A. 2. min A A,, inf A = min A.. M = max A. A U(A). (ii) M A, *i M U(A). (i) M A. *ii, A m U(A) M m, M U(A). M = min U(A), A..

15, 73 *i A. *ii A. 15.7. a, b R, a < b. sup(a, b) = b, inf(a, b) = a.. 14.5, U((a, b)) = [b, ), L((a, b)) = (, a]., sup(a, b) = min U((a, b)) = min[b, ) = b inf(a, b) = max L((a, b)) = max(, a] = a. 15.8 ([8, p.58]). A R, s R.. 1. 2. s = sup A s = inf A { (i) (ii) { (i) (ii) x A : x s r < s, x A : r < x x A : x s r > s, x A : r > x. A U(A). (i) s A, s U(A). (ii) r < s, r A., r A, s r, s U(A). *i (i),(ii) s U(A), sup A. *i (ii) r < s, x A : r < x r R : (r < s) ( x A : r < x) r R : ( x A : r < x) r s r R : ( x A : r x) r s (ii) r R : ( x A : x r) s r r R : r U(A) s r r U(A) : s r

74., s U(A).., r A x A : r < x.. 15.9. A R, s R.. 1. 2. s = sup A s = inf A { (i) (ii) { (i) (ii) x A : x s ε > 0, x A : s ε < x x A : x s ε > 0, x A : s + ε > x 15.10. a, b R, a < b. sup(a, b) = b 15.8.. x (a, b), a < x < b, x b, b 15.8 (i). (ii). r < b. x = max{(a + b)/2, (r + b)/2}, x (r + b)/2 > r x > r., x (a + b)/2 > a x > a., (a + b)/2, (r + b)/2 < b x < b., x (a, b) r < x. (ii). b = sup(a, b)., 14.5., r a, x = (a + b)/2. a < r, x = (r + b)/2. inf(a, b) = a. 15.11., r b, r (a, b). r,, r > a. 15.2 R. 15.12 ( (Archimedes) [8, p.65]). R. 1. N. 2. a > 0, b R, n N : na > b. 3. ε > 0, n N : 1 n < ε.,.

15, 75. N. N. 15.2, N. s = sup N. 15.9, n N, s 1 < n. s < n + 1, n + 1 N, s N. N.,. *i, 1,2,3. N, x R, n N : n > x. 1 2.) a, b R, a > 0., n N, n > b/a. a > 0, a, na > b. 2 3.) ε > 0., n N, nε > 1. n > 0, n, ε > 1/n. 3 1.) x > 0. *ii 1/x > 0,, n N 1/x > 1/n. nx > 0, nx, n > x. *i N, 15.2, N. 15.8, N, ` s R : ( N N : N s) ( r < s, n N : r < n) s R : ( N N : N s) ( r < s, n N : r < n) s R : ( r < s, n N : r < n) ( N N : N s) s R : ( r < s, n N : r < n) ( N N : N s),, s R : ( r < s, n N : r < n) ( N N : N > s) s R : ( N N : N s) ( r < s, n N : r n)... s N, s N, s N., N, s., s,, s, s, s. *ii x,, x.

76 15.13.,.,,.,,. 15.14. E = { 1 n n N } R., max E = sup E = 1, inf E = 0, min E.. max E = 1. (i) 1 N, 1 = 1/1 E. (ii) n N, n 1, 1/n 1. max E = 1. max E = 1, sup E = 1 15.6. inf E = 0. 15.8 15.9. (i) n N, n > 0, 1/n > 0. (ii), ε > 0, n N, ε > 1/n., inf E = 0. inf E = 0 E, 15.6, min E. *i,, n N, 1/n > 1/(n + 1) E. *ii, min E. *i min E, inf E = min E E. *ii min E, m E, x E : m > x. 15.15., E (0, 1], E = (0, 1]. 4.10. 15.16. 15.8. 15.17. = A B R, B. 1. A. 2. sup A sup B. 3. inf A inf B.

15, 77 15.18. A R. 1. r R A., x r, x A. 2. m R A., x m, x A. { 15.19. E = ( 1) n n 2n + 1. } n N R. max E, sup E, min E, inf E

78 16 16.1 ([8, p.93]). N R a: N R,,. a(n) a n, {a n } n N {a n }. 16.2 ([8, pp.94 95]). {a n } α R ε > 0, N, n N n N def, a n α < ε. *i, ε > 0, N N, n(n N) : a n α < ε. lim a n = α n a n α (n ). α {a n }. {a n } α R.. *i., ε > 0, N N, n N : n N a n α < ε. ε > 0, N, n N a n α < ε.,,. 16.3. ε-n. 16.4. {a n } ε-n.

16 79 {a n },, {a n }. {a n }, α R, ε > 0, N N, n(n N) : a n α < ε., α R, ε > 0, N N, n(n N) : a n α ε.,. α, ε, N, N n, a n α ε. 16.17, 16.18 16.5 ([8, p.95]). 1. {a n } def K R, N N, n N a n > K.,, lim n a n = +. 2. {a n } K R, N N, n(n N) : a n > K. def K R, N N, n N a n < K.,, lim n a n =. K R, N N, n(n N) : a n < K. 16.6.,,.,. 16.18 {a n } α, lim n a n = α, α, {a n },,,. 16.7 ([8, p.108]). {a n },,.

80. α R {a n }. β R, α β. β {a n } *i. ε = α β /2 ε > 0. x R x α < ε., α β = α x + x β α x + x β < ε + x β, x β > α β ε = ε. α = lim n a n, ε, N 1, n N 1 a n α < ε., n N 1 a n β > ε, β {a n } *ii. *i ε > 0, N N, n(n N) : a n β ε *ii N, n := max{n, N 1 }, n N, n N 1, a n β ε. 1 16.8. 1. lim n n = 0. 2. lim n = +. n. 1. ε > 0, 15.12, N N, 1/N < ε. n N n N, 1/n 1/N, 1 n 0 = 1 n < ε 1, lim n n = 0. 2. K R, 15.12 N, N N, N > K. n N n > K, lim n n = +..

16 81 16.9 ([8, p.110]). {a n }, {b n } {a n + b n }, {a n b n }, {a n b n } 1. lim (a n + b n ) = lim a n + lim b n, lim (a n b n ) = lim a n lim b n. n ( n ) ( n ) n n n 2. lim (a nb n ) = lim a n lim b n. n n n k R, {ka n }, lim ka n = k lim a n. n n b n 0 (n = 1, 2,... ) lim n b n 0 {a n /b n } a n 3. lim = n b n lim n a n lim n b n. α = lim n a n, β = lim n b n. 1. ε > 0. lim a n = α, N 1 N, n N 1 n N n, a n α < ε 2., lim n b n = β, N 2 N, n N 2 n N, b n β < ε 2. N = max{n 1, N 2 }., n N n N, n N 1 n N 2,, lim n (a n ± b n ) = α ± β. 2. ε > 0. (a n ± b n ) (α ± β) = (a n α) ± (b n β) { ε := min a n α + b n β < ε 2 + ε 2 = ε. ε 1 + α + β, 1, ε *i > 0. lim a n = α, ε, n N 1 N, n N 1 n N, a n α < ε., lim n b n = β, N 2 N, n N 2 n N, b n β < ε. }

82 N = max{n 1, N 2 }., n N n N, n N 1 n N 2, a n b n αβ = (a n α + α)(b n β + β) αβ = (a n α)(b n β) + α(b n β) + β(a n α) a n α b n β + α b n β + β a n α < ε ε + α ε + β ε 1 ε + α ε + β ε = (1 + α + β )ε (1 + α + β ) = ε. ε 1 + α + β, lim n a nb n = αβ. b n = k,. *i,, a n α, b n β, a n b n αβ., a n b n αβ, a n α b n β, a n b n αβ = (a n α)(b n β) + α(b n β) + β(a n α)., a n α, b n β ε, a n b n αβ (ε + α + β )ε. ε ε.,.. ε > 0. j ε := min ε 1 + α, ε 1 + β, 1, ε > 0. lim a n = α, ε, N 1 N, n n N 1 n N, a n α < ε 2., lim b n = β, N 2 N, n N 2 n N n, b n β < ε 2. N = max{n 1, N 2 }., n N n N, n N 1 n N 2, ff

16 83 a n b n αβ = a n b n a n β + a n β αβ a n b n a n β + a n β αβ = a n b n β + a n α β = a n α + α b n β + a n α β ( a n α + α ) b n β + a n α (1 + β ) (1 + α ) b n β + a n α (1 + β ) ( a n α < ε /2 1.) < (1 + α ) ε 2 + ε (1 + β ) 2 ε 2 + ε 2 = ε... 16.10. {a n }, {b n }, α = lim a n, β = lim b n. {c n } n n.. 1. N, n N a n b n., α β. 2. α = β., N, n N a n c n b n., {c n } lim n c n = α. 1., α > β, N, n N, a n > b n *i. α > β. ε = α β ε > 0. 16.9.1 *ii {a n b n } α β, N 0, m N 0 (a m b m ) (α β) < ε,, ε < a m b m (α β) < ε. ε = α β,, m N 0 *iii. a m b m > α β ε = 0

84 N, n := max{n, N 0 }, n N. n N 0, a n > b n *iv.,, 13.12. N, n N a n b n., α β, α β 0. 13.12, ε > 0, α β ε. ε > 0. 16.9.1 {a n b n } α β, N 0, m N 0 (a m b m ) (α β) < ε,, ε < a m b m (α β) < ε., m N 0 α β < a m b m + ε. n := max{n, N 0 }, n N, a n b n 0., n N 0, α β < a n b n + ε., α β < a n b n + ε 0 + ε = ε.,. *i ( N N, n N : n N a n b n ) α β α β ( N N, n N : n N a n b n ) α > β ( N N, n N : n N a n > b n ). *ii ε = α β, 16.9.1. *iii 2.,. *iv, α > β ( N 0 N, m(m N 0 ) : a m > b m ) ( N 0 N, m(m N 0 ) : a m b m ) α β ( N 0 N, m(m N 0 ) : a m > b m ) ( N N, n N : n N a n > b n ).,.

16 85 2. ε > 0. lim a n = α, N 1 N, n N 1 n N n, a n α < ε, ε < a n α., lim n b n = α, N 2 N, n N 2 n N, b n α < ε, b n α < ε. N 0 = max{n, N 1, N 2 }., n N 0 n N, n N n N 1 n N 2, a n α ε < n N 1 n N c n α n N b n α < ε, n N 2, c n α < ε. 16.11. 16.10.2, {c n }. {c n }, 16.10.1, a n c n c n b n,, {c n }.,,. 16.12 ([8, p.121]). {a n n N}. def 2. {a n } {a n n N}. def 3. {a n } 1. {a n } {a n n N}. def 4. {a n n N}, {a n }, sup a n. 5. {a n n N}, {a n }, inf a n. 16.13. {a n }, 1. n N a n a n+1. 2. n N a n < a n+1. 3. n N a n a n+1. 4. n N a n > a n+1. 5...,.

86 16.14. {a n }... lim n a n, sup a n. {a n } α, α = sup a n. N N. {b n } b n = a N lim n b n = a N. {a n }, n N a n a N = b n. Thm. 16.10.1 α = lim a n lim b n = a N. N N α a N, α {a n } n n. b {a n }. n N a n b. Thm. 16.10.1 α = lim n a n b. α {a n }, α = sup a n. sup a n α. ε > 0, N a N > α ε 15.9. {a n } n N a n a N., α {a n } n N α a n., n N α ε < a n α, a n α < ε. {a n } a. 16.15. 1.,. 2.,.,. 16.16. A R., A {a n }, n N : a n A, lim n a n = sup A.. n N, 1/n > 0, 15.9, A x, sup A 1 n < x. n N, x A, a n. {a n }, n N, a n A,.. sup A 1 n < a n sup A lim (sup A 1 n n ) = sup A,, lim a n = sup A n 16.17. {( 1) n }. 16.18. 1. {a n },

16 87. 2. {a n } {a n },. 3. lim n a n = lim n ( a n) = +. 16.19.. 16.20. {a n }, n N : a n < a n+1 α R., n N, a n < α.

88 17 17.1 ([8, p.141]). X. 1. f : X R, X. 2. f : X R. f(x) = {f(x) x X}, f., R.,,,,. 17.2 ([8, p.142], [5, p.16]). S( ) R. a R S ε > 0, x S : 0 < x a < ε. def 17.3. a R S, a a S. a S. 0 (0, 1). S R, S, S,. 17.4 ([8, p.142]). S( ) R, a R S, α R, f : S R., f(x) x a α def ε > 0, δ > 0, 0 < x a < δ x S, f(x) α < ε., ε > 0, δ > 0, x S(0 < x a < δ) : f(x) α < ε. lim f(x) = α x a f(x) α (x a). α, x a f.

17 89 17.5., x a f, a f. 17.6. ε-δ. 17.7. x a f, x a, x = a f. f a., 17.4 0 < x a < δ 0 <.. 17.8. f : R R f(x) = { 1, x 0 0, x = 0., x 0, f(x) 1, lim f(x) = 1 x 0.. ε > 0., δ = 1 *i, 0 < x 0 < δ x R, x 0 f(x) = 1, f(x) 1 = 1 1 = 0 < ε., ε > 0, δ > 0, x R(0 < x 0 < δ) : f(x) 1 < ε., f, lim f(x) = 1 x 0 ε > 0, δ > 0, x R( x 0 < δ) : f(x) 1 < ε *ii., ε = 1 *iii., δ > 0, 0 0 = 0 < δ, f(0) 1 = 0 1 = 1 ε. *i. *ii,. *iii 0 < ε 1. ε > 0, δ > 0, x R( x 0 < δ) : f(x) 1 ε

90 17.9. f(x) x a α. f(x) x a α ε > 0, δ > 0, x S(0 < x a < δ) : f(x) α < ε. ε > 0, δ > 0, x S(0 < x a < δ) : f(x) α < ε ε > 0, δ > 0, x S : 0 < x a < δ f(x) α ε. ε > 0, δ > 0, x S, 0 < x a < δ f(x) α ε. *i *i, a, f(x) α, x S. 16.7, 16.9, 16.10... 17.10 ( 16.7 ). x a f,,.. S f, α R, x a f. β R, α β. β *i. ε = α β /2 ε > 0. y R y α < ε., α β = α y + y β α y + y β < ε + y β, y β > α β ε = ε. α = lim x a f(x), ε, δ 1 > 0, 0 < x a < δ 1 f(x) α < ε., 0 < x a < δ 1 f(x) β > ε. *ii

17 91 δ > 0, min{δ, δ 1 } > 0, x S 0 < x a < min{δ, δ 1 } a S. x S, 0 < x a < δ, 0 < x a < δ 1 f(x) β ε., β. *i ε > 0, δ > 0, x S : 0 < x a < δ f(x) β ε. 17.9. *ii,.,.. 17.11 ([8, p.147], 16.9 ). f : S R, g : S R, lim f(x) = α, lim g(x) = β., α, β R. x a x a, lim (f(x) + g(x)), lim (f(x) g(x)), lim (f(x)g(x)), x a x a x a. 1. lim (f(x) + g(x)) = α + β, lim (f(x) g(x)) = α β. x a x a 2. lim (f(x)g(x)) = αβ. x a k R, lim kf(x), x a lim kf(x) = kα. x a x S : g(x) 0 β 0 lim x a f(x) g(x), 3. lim x a f(x) g(x) = α β.. 1. ε > 0. lim x a f(x) = α, δ 1 > 0, 0 < x a < δ 1 x S, f(x) α < ε 2., lim x a g(x) = β, δ 2 > 0, 0 < x a < δ 2 x S, g(x) β < ε 2. δ = min{δ 1, δ 2 } δ > 0, 0 < x a < δ x S, 0 < x a < δ 1 0 < x a < δ 2,

92 (f(x) ± g(x)) (α ± β) = (f(x) α) ± (g(x) β), lim (f(x) ± g(x)) = α ± β. n 2. ε > 0. { ε := min f(x) α + g(x) β < ε 2 + ε 2 = ε. ε 1 + α + β, 1, ε > 0. lim x a f(x) = α, ε, δ 1 > 0, 0 < x a < δ 1 x S, f(x) α < ε., lim x a g(x) = β, δ 2 > 0, 0 < x a < δ 2 x S, g(x) β < ε. δ = min{δ 1, δ 2 }., 0 < x a < δ x S, 0 < x a < δ 1 0 < x a < δ 2, f(x)g(x) αβ = (f(x) α)(g(x) β) + α(g(x) β) + β(f(x) α) f(x) α g(x) β + α g(x) β + β f(x) α < 1 ε + α ε + β ε = (1 + α + β )ε (1 + α + β ) = ε. ε 1 + α + β, lim x a (f(x)g(x)) = αβ. g(x) = k,... 17.12 ( 16.10 ). f : S R, g : S R, α = lim x a f(x), β = lim x a g(x)., α, β R. h: S R. 1. δ 0 > 0, 0 < x a < δ 0 f(x) g(x)., α β. }

17 93 2. α = β., δ 0 > 0, 0 < x a < δ 0 f(x) h(x) g(x)., lim h(x), lim h(x) = α x a x a.. 1., α > β, δ > 0, 0 < x a < δ x S, f(x) > g(x). α > β. ε = α β ε > 0. f(x) g(x) x a α β, δ 0, 0 < x a < δ 0 x S (f(x) g(x)) (α β) < ε,, ε < f(x) g(x) (α β) < ε. ε = α β,, 0 < x a < δ 0 x S. f(x) g(x) > α β ε = 0 δ > 0, min{δ, δ 0 } > 0, 0 < x a < min{δ, δ 0 } x S. x S, 0 < x a < δ,, 0 < x a < δ 0, f(x) > g(x). 2. ε > 0. lim x a f(x) = α, δ 1 > 0, 0 < x a < δ 1 x S, f(x) α < ε, ε < f(x) α., lim x a g(x) = α, δ 2 > 0, 0 < x a < δ 2 x S, g(x) α < ε, g(x) α < ε. δ = min{δ 0, δ 1, δ 2 }., 0 < x a < δ x S, 0 < x a < δ 0 0 < x a < δ 1 0 < x a < δ 2, ε < f(x) α h(x) α g(x) α < 0< x a <δ 1 0< x a <δ 0 0< x a <δ 0, h(x) α < ε. 0< x a <δ 2 ε,

94 18 18.1 ([8, p.154]). S( ) R, f : S R. 1. a S. f x = a continuous lim f(x) = f(a). def x a 2. f S a S, f x = a. def 18.2., lim x a f(x), a a f, x = a f f(a). f x = a,, x = a f. 18.3. f x = a, ε > 0, δ > 0, x S( x a < δ) : f(x) f(a) < ε (3).,, f x = a, lim x a f(x) = f(a).,. ε > 0, δ > 0, x S(0 < x a < δ) : f(x) f(a) < ε (4) (3), (4). x 0 < x a < δ, x a < δ., x a = 0 x, x = a, ε > 0, f(x) f(a) = f(a) f(a) = 0 < ε, (4), (3). f x = a, (4),, (3), (3). 18.4. 17.8 f : R R f(x) = { 1, x 0 0, x = 0. f, x = 0., a 0, x = a.

18 95, lim x 0 f(x) = 1 0 = f(0), x = 0., a 0. ε > 0, δ = a, δ > 0. x a < δ x, a = a x + x a x + x < δ + x = a + x, x > 0, x 0.,, f x = a. f(x) f(a) = 1 1 = 0 < ε. 18.5. S( ) R, a S, f : S R.. 1. f x = a. 2. a, S {a n },. lim f(a n) = f(a) n. 1 2. f x = a. {a n } S, lim a n = a., lim f(a n) = f(a) *i n n. ε > 0. f x = a, ε, δ > 0, x a < δ x S, f(x) f(a) < ε. lim n a n = a, δ, N N, n N n N, a n a < δ., n N n N, f(a n ) f(a) < ε. *i, ε > 0, N N, n(n N) : f(a n ) f(a) < ε.

96. *i,, f x = a *ii, a, S {a n }, {f(a n )} f(a) *iii *iv. f x = a., ε > 0, δ > 0, x a < δ x S, f(x) f(a) ε. n N, 1/n > 0, x a < 1/n x S, f(x) f(a) ε. n N, x S, a n. {a n }.,, n N, a n S., n N, 0 a n a < 1/n,, lim n a n = a., N N, f(a N ) f(a) ε., {a n }, S a,, {f(a n )} f(a). *i, *ii 17.9, ε > 0, δ > 0, x S( x a < δ) : f(x) f(a) ε.. 18.3. *iii, ε > 0, N N, n(n N) : f(a n ) f(a) ε. *iv {a n }(( n N : a n S) lim a n = a) : lim f(a n) = f(a) n n {a n }(( n N : a n S) lim n a n = a) : {f(a n )} {f(a)}.,. 18.6. f : [a, b] R, f([a, b]),.

18 97 18.7.,. [5].,,,. 18.8 ( ). f : [a, b] R, f(a) f(b)., f(a) f(b) k, f(c) = k a < c < b c.. 1. f(a) < f(b). f(a) < k < f(b), E := {x [a, b] f(x) k}. E [a, b] E, a E E., E. c := sup E. f(b) y k f(a) a E c b x f(c) = k. 13.15, ε > 0, f(c) k < ε,, k ε < f(c) < k + ε.

98 ε > 0. f, δ > 0, x c < δ x [a, b], f(c) f(x) < ε,, f(x) ε < f(c) < f(x) + ε. c = sup E, x 0 E, c δ < x 0 15.9. x 0 E, c = sup E, x 0 c. x 0 c < δ. f(c) < f(x 0 ) + ε. x 0 E, f(x 0 ) k., f(c) < k + ε., f(c) < f(b) *i c b., c < b. δ 0 := min{δ, b c}/2, c + δ 0 [a, b], c + δ 0 c = δ 0 < δ, f(c + δ 0 ) ε < f(c). c + δ 0 > c = sup E, c + δ 0 E., k < f(c + δ 0 )., k ε < f(c)., f(c) = k. f(c) = k f(a), a c. a < c < b. 2. f(a) > f(b). f(a) > k > f(b). g : [a, b] R, g(x) := f(x)., g *ii g(a) < k < g(b).,, g(c) = k a < c < b c. f(c) = g(c) = k, c. *i ε f(b) k.,, f(c) k. *ii. f(a) < f(b). f(a) < k < f(b), E, c. f(c) k. c = sup E, 16.16, E {a n } c.

18 99 n N, a n E, f(a n ) k., f, 18.5, lim n f(a n) = f(c)., 16.10.1, f(c) k. f(c) k. f(c) k < f(b), c b., c < b. {b n } b n = c + 1 n. lim n b n = c. b c > 0, N N, 1 N < b c. n N n N, a c < b n = c + 1 n c + 1 N < b, b n [a, b]., b n > c = sup E, b n E., n N, f(b n ) > k., f, lim n f(b n) = f(c) *i., 16.10.1, f(c) k., f(c) = k. a < c < b. *i, n < N b n [a, b], f(b n )., {f(b n )} n N,, b n [a, b] n f(b n ). 18.1., f : [a, b] R (a, b),,..,. 18.9. f : R R f(x) = { x, x 0 1, x = 0. 1. f, x = 0. 2. f, a 0, x = a.

100 18.10. f : R R x = a, f(a) > 0., a f(x),, δ > 0, x a < δ x R, f(x) > 0. 18.11. {a n } α R., N N, n(n N) : a n > 0. 18.12. {a n }, {b n }, lim a n = α, lim b n = β., n n n N, a n < b n., α < β,.

101 [1] H.D.,.., 2004. [2]. -., 2009. [3].. http://math.u-ryukyu.ac.jp/~tsukuda/ lecturenotes/. [4].., 2008. [5],.., 1996. [6].., 2002. [7].., 1994. [8].., 1999. [9].. http://math.u-ryukyu.ac.jp/ ~tsukuda/lecturenotes/.