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funksional analiz misol va masalalar yechish 1 qism
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∈ A∪B. ' y = f(x) ∈ f(A∪B) f (A) ∪ f(B) ⊂ f(A ∪ B). ' 3%%5 ! () * +, * *+- + # - %$"%##" & . Z R : R → R, R(x) = ⎧ ⎨ ⎩ 1 n , agar x = m n , m ∈ Z, n ∈ N 0, agar x ∈ R\Q. (2.3) ' m n − / . & P : R 2 → R, P (x, y) = x. 2 S : R 3 → R, S(x 1 , x 2 , x 3 ) = x 2 1 + x 2 2 + x 2 3 . %%" g A = [0, 3) & B = (1, 4) & & %$" R A = R\Q & B = (1, ∞) & & 2 3 5 3 " 5 ' $ ' " ' % * f α ∈ R ! α f & * f α ∈ R ! α + f ' * f : R → R α ∈ R ! α f . * f : R → R α ∈ R ! α + f / * f β ∈ R α ∈ R ! α f + β * f (x) = k x + l, k > 0 ! f (a) = c, f(b) = d f : [a, b] → [c, d] ! & ( f −1 (A ∪ B) = f −1 (A) ∪ f −1 (B). & ( f −1 (A ∩ B) = f −1 (A) ∩ f −1 (B). < - f −1 ∪ α A α = ∪ α f −1 (A α ), f −1 ∩ α A α = ∩ α f −1 (A α ). $ 3%`5 3! ! 5 & ! ( f ∪ α A α = ∪ α f (A α ) % , & ' & %a" & P (x, y) = x A = {(x, y) : 0 ≤ x ≤ 1, y = 0}, B = {(x, y) : 0 ≤ x ≤ 1, y = 1} & P (A ∩ B) = P (A) ∩ P (B) c ' f (A ∩ B) ⊂ f (A) ∩ f (B) . ' X & A ⊂ X & X & χ A (x) = ⎧ ⎨ ⎩ 1, agar x ∈ A 0, agar x / ∈ A (2.4) A " X \A; A ∪ B; A ∩ B; A\B; AΔB & χ A (x) χ B (x) / < χ ∪ α A α (x) = sup α χ A α (x) ; χ ∩ α A α (x) = inf α χ A α (x) . f (x) = x 2 , - 5 f : R → R 4 5 f : R → R + 4 !5 f : R + → R + * f : A → B g : B → A + ϕ : A → B + * f : A → B g : B → A + A B + c f : R 2 → R f ((x, y)) = y A = {(0, y) : y ∈ R} B = {(1, y) : y ∈ R} & ! f (A ∩ B) f (A) ∩ f (B) & & ' f : R 2 → R ! f (A ∩ B) e f (A) ∩ f (B) c $ f : X → Y A, B ⊂ X & ! A ⊂ B f (A) ⊂ f (B) % f : X → Y ! + ! - 5 f − 4 5 A, B ⊂ X ! f (A ∩ B) = f (A) ∩ f (B) 4 !5 ! B ⊂ A & ! f (A\B) = f (A) \f (B) 4 5 A ⊂ X ! f −1 (f (A)) = A & f : X → Y A, B ⊂ Y & ! 5 f f −1 (A) = A ∩ f (X) 5 A ⊃ B f −1 (A\B) = f −1 (A) \f −1 (B) " ' f : R → R f (x) = 0, 5 · [x] * A = [0, 8], B = (2, 3) f (A) f −1 (B) & & . f : X → [5, 10], f(x) = x 2 +1 f 3 5 X & & / f : X → R + , f (x) = x 2 + 1 X & f c M (x, y) (0, 1) × (0, 1) , x y ! " x = 0, n 1 n 2 n 3 . . . y = 0, m 1 m 2 m 3 . . . < f : (0, 1) × (0, 1) → (0, 1) f (M(0, n 1 n 2 n 3 . . . , 0, m 1 m 2 m 3 . . . )) = P (0, n 1 m 1 n 2 m 2 n 3 m 3 . . . ) ' c ' ! c f : [0, π] → [−1, 1], f (x) = cos x, g : [0, π] → [0, 1], g (x) = sin x, ϕ : [0, π 2 ] → [0, 1], ϕ (x) = sin x, ψ : [0, 3] → [0, 10], ψ (x) = x 2 + 1, ! + § 2** - -34 5 60** "- 0 - = & & ( & + r + ' & + ' & : + 1 & : + " * a b # ! : + ! a b # ! b c # ! a c # ' a b b c , # a c " : ( : + [ & ! " : + ! 7, X × X R (a, b) ∈ R a b R a ∼ b R 7, ! R " # # a ∈ X a ∼ a R $ % &' % ! a ∼ b R b ∼ a R $ &' ` ! a ∼ b R b ∼ c R a ∼ c R $ & 60** "- "0 * # "0+ d " & & d & ! ! " 7, ! M M ' ! M & " a 1 , a 2 , . . . , a n , . . . ! " ! M d & " {} # % ` b & {1, 2, 3, 4} 7, ( $7, ! A B A ∼ B [ & ! ! (: - &" & %7, [0, 1] "" [0; 1] = f' & ' ! "" $) *+ & ! A B B 1 B A A 1 A B 2** * ! & " * A B & 2 " & ! A & A * A B & Download 1.57 Mb. Do'stlaringiz bilan baham: 
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