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E\A
α ) (1.1j) ! [ * x ∈ α (E\A α ) α 0 ! x ∈ E\A α 0 x A α 0 & x ∈ α A α \ x ∈ E\ α A α ' E \ α A α ⊃ α (E\A α ) (1.2j) 3##+5 3#%+5 3#b5 ' A ∪ B = [0, 1], A\B = {0, 1} , AΔB = {0, 1} , A ∩ B = (0, 1). . A ∪ B = {2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, . . .} , A \B = {2, 4, . . . , 2(3n − 2), 2(3n − 1), . . .} , A ∩ B = {6, 12, . . . , 6n, . . .} A ΔB = {2, 3, 4, 8, 9, 10, 14, 15, 16, . . .} . / A ∪ B = R, A\B = Q A ΔB = R, A ∩ B = ∅. A ∪ B = [0, 1], A\B = A A ΔB = [0, 1], A ∩ B = ∅. A = πn 4 , n ∈ Z , B = π 4 + πn 2 , n ∈ Z ' B ⊂ A " A ∪ B = A, A ∩ B = B, A\B = AΔB = πn 2 , n ∈ Z A & ##"! B & ##" ! A ∪B = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} , A \B = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y < x} , A ∩ B = {(x, y) : 0 ≤ x ≤ 1, y = x} , A ΔB = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y < x} ∪ {(x, y) : 0 ≤ x ≤ 1, x < y ≤ 1} . ""# A ∪ B − #%"! A \B − ! " A ΔB; − ! A ∩ B − ! "&# . Z = Z, X = {1, 2}, Y = {3} X × Y = Y × X X = Y ! / X × Y = {(1, 2), (1, 4), (3, 2), (3, 4), (5, 2), (5, 4)}, Y × X = {(2, 1), (2, 3), (2, 5), (4, 1), (4, 3), (4, 5)}. X × Y = Y × X 1 X = [0, 1], X 1 = [1, 2], Y = [2, 3], Y 1 = [3, 4]. . A ∗ = B ∪ C ∪ D, A ∗ = B ∩ C ∩ D. $ x m = (−1) m " Ω {x m } /∈ ∞ n =1 Ω n . § 0- . E (R) = 0, 1 2 , 1 3 , . . . , 1 n , . . . . / E (P ) = R. E (S) = [0, ∞). g (A) = {0, 1, 2}, g −1 (B) = [2, 4). R(A) = {0}, R −1 (B) = ∅. f (x) = x, g(x) = 1 − x, (f + g)(x) = 1 f (x) = x, g(x) = x − 3, (f − g)(x) = 3. % %%8" & A B & ( A ∩ B = ∅. ' " P (A ∩ B) = ∅ ' & P " ( P (A) = [0, 1] P (B) = [0, 1] \ P (A) ∩ P (B) = [0, 1] = ∅ = P (A ∩ B) . χ X \A (x) = 1 − χ A (x) , χ A ∪B (x) = χ A (x) + χ B (x) − χ A (x) · χ B (x) , χ A ∩B (x) = χ A (x) · χ B (x) , χ A \B (x) = χ A (x) − χ A (x) χ B (x) χ A ΔB (x) = χ A (x) + χ B (x) − 2χ A (x) · χ B (x) + 1 ' f (A) = 1 2 , 1, 3 2 , 2, 5 2 , 3, 7 2 , 4 f −1 (B) = [5, 6) . X = [−3, −2] ∪ [2, 3] / X ⊂ (−∞, 0] X ⊂ [0, +∞) & ' f, ϕ, ψ. 2 f, g, ϕ. ' f, ϕ. §. 2** ' K a = {(x, y) : x = a, −∞ < y < ∞} − x = a ! . K r = (x, y, z) : x 2 + y 2 + z 2 = r − " r ≥ 0 ' f (2n) = n . ' & f : R → (0, 1), f(x) = 1 π arctg x + 1 2 ' ! _ ! & A = N, B = {3, 4, . . . , n, . . .} . A = N, B = {. . . − 2, −1, 0, 1, 2}. % 2 A = 0, 1, 1 2 , 1 3 , . . . , 1 n , . . . , B = 1 2 , 1 3 , . . . , 1 n , . . . & , [0, 1]\A = (0, 1)\B = C A B & ! f : A → B + , g (x) = ⎧ ⎨ ⎩ f (x), agar x ∈ A x, agar x ∈ C [0, 1] (0, 1) f (x, y) = a + x + 1 2 (b − a), c + y + 1 2 (d − c) (x, y) ∈ [−1, 1] × [−1, 1]. % A ⊂ R\Q & B = A ∪ Q & f : A → B + , g (x) = ⎧ ⎨ ⎩ f (x), agar x ∈ A x, agar x ∈ R\(Q ∪ A) R\Q R " . [ $ ! $. A = [0; 1] ! `b" ! & & & %/ A ⊂ [0, 1] & f (x) = χ A (x) − A & , χ A A & \ [0, 1] 0 1 ! & [0, 1] ! & + [0, 1] & ! & & [0, 1] 0 1 ! & & & % `b" R 2 ! & & & % ' & 7>" & ! & § " / 2 & & $ * A, B ∈ P = ∩ α P α α A, B ∈ P α P α ! A ΔB ∈ P α , A ∩ B ∈ P α . , A ΔB ∈ P A ∩ B ∈ P. _ ' $ _ % / 2 & A n = n −1 ∈ S ∞ n=1 A n / ∈ S. S Oxy & S 45 Oxy 45 0 & " S ∪ S 45 d & & S 1 = {A, ∅} , S 2 = {A, {a}, {b, c}, ∅} , S 3 = {A, {b}, {a, c}, ∅} , S 4 = {A, {c}, {a, b}, ∅} , S 5 = {A, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, ∅} " P 1 = {∅, {a}} , P 2 = {∅, {b}} , P 3 = {∅, {c}} ' & A = [0, 7], B = (1, 3) ! A ΔB = [0, 1] ∪ [3, 7] & $ A = (0, 7] × (0, 7] B = (0, 5] × (3, 5] & ! A ΔB % ' + ' & [ b ! & a> ' ' σ − σ − . P A B 1 = A∩B 1 , B 2 = A ∩ B 2 , B 1 ∩ B 2 = (A ∩ B 1 ) ∩ (A ∩ B 2 ) = A ∩ (B 1 ∩ B 2 ) ∈ P A . B 1 Δ B 2 = (A ∩ B 1 )Δ(A ∩ B 2 ) = ((A ∩ B 1 ) ∪ (A ∩ B 2 )) \ (A ∩ (B 1 ∩ B 2 )) = = A ∩ (B 1 ΔB 2 ) ∈ P A \ P A ' A & ( P A * P σ − P A & σ − / X & 5 _ 5 !5 $ §. # * ' A = 1 27 , 2 [3, 4) , μ(A) = 2 26 27 . . A = 1 27 , e 3, e 2 ∪ 9, e 3 , μ (A) = e 3 + e 2 + e − 12 1 27 . / A = 1 60 , 1 24 1 16 , 5 2 , μ (A) = 2 37 80 . A = 3 256 , 1 4 1 256 , μ (A) = 61 256 . A = 0, 3 8 3 8 , 13 24 7 8 , 25 24 , μ (A) = 17 24 . & μ (A) = 0. ' μ (A) = 1 2 . . μ (A) = 1. / A k ⊂ [0, 1], k ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9} ! k & ' ! 5 " & ! # ab" k ∈ {1, 2, . . . , 9} ! μ (A k ) = 1 2 ! # $ ! ! " & A 1 ∩ A 2 ∩ · · · ∩ A 9 , μ [0, 1] \ ( 9 k =1 A k ) = μ 9 k =1 ([0, 1] \ A k ) ≤ 9 k =1 μ ([0, 1] \ A k ) = 0 \ μ (A 1 ∩ A 2 ∩ · · · ∩ A 9 ) = 1 2 A = [0, 1]∩Q μ (A) = 0, μ(A) = 1 ' 5 b`" ( Download 1.57 Mb. Do'stlaringiz bilan baham: |
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