WebNov 4, 2015 · 1 Answer. Sorted by: 2. Recall that f ( − 1) ( f ( A)) = { x ∈ X ∣ f ( x) ∈ f ( A) } and that f ( A) = { f ( x) ∣ x ∈ A }. If x ∈ A, then f ( x) ∈ f ( A) by definition and then x ∈ f ( − 1) ( f ( A)). Share. Cite. Follow. WebShow that it is always true that f (A ∪ B) = f (A) ∪ f (B) and f (A ∩ B) ⊂ f (A) ∩ f (B). (a) Show, by example, that it might happen that f (A ∩ B) # f (A) ∩ f (B) (b) Show that f (A ∩ …

For which of the following functions is f(a+b)=f(b) + f(a)

Web1.2.22 (a) Prove that f(A ∩ B) = f(A) ∩ f(B) for all A,B ⊆ X iff f is injective. Proof. We show the implications separately. =⇒: Let x 1,x 2 ∈ X be arbitrary with f(x 1) = f(x 2). Let A = {x … WebMay 13, 2024 · f ( a + b) = f ( a) + f ( b) For a = b we have f ( 2 a) = 2 f ( a) Taking a = 0 gives you f ( 0) = 0 and for b = − a we have f ( 0) = 0 = f ( x) + f ( − x) so f is odd. Then you can show by induction that f ( n) = n f ( 1) Then f is odd so it is true for n ∈ Z. With r … fau walsall manor

Classical Analysis I 1 Sets, relations, functions

WebExercise 3. Let F = [a 1 , b 1 ] × … × [a n , b n ] ⊂ R n and let ϵ > 0; use Exercise 2 to show that there are rectangles R 1 , …, R m such that F = k = 1 ⋃ m R k and diam R k < ϵ for … Webf(A) is called the image of f, and is denoted Im(f). Given T ⊂ B we define f−1(T) = {a ∈ A f(a) ∈ T}. Note that f−1(T) ⊆ A. f−1(T) is called the preimage of the set T under f. Fix a … Web[F(a): F] = n, n is odd then it is clear F ⊂ F(a2) ⊂ F(a) (because a2 ∈ F(a) ). So (by law, or Theorem): [F(a): F] = [F(a): F(a2)][F(a2): F] If prove [F(a): F(a2)] = 1 then F(a) = F(a2). Conversely, suppose [F(a): F(a2)] ≠ 1. Since [F(a): F(a2)] can’t be even (because n is odd) then [F(a): F(a2)] ≥ 3 . friedman and son