TOPOLOGY Let f : X → Y be a function. Prove that f is one-to-one and...

TOPOLOGY

Let f : X → Y be a function.

Prove that f is one-to-one and onto if and only if f[A^c] = (f[A])^c for every subset A of X. (prove both directions)

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Colby Messinger answered 8 months ago

Topology question: Prove that a bijection f : X → Y is a homeomorphism if and...

Topology question: Prove that a bijection f : X → Y is a homeomorphism if and only if f and f−1 map closed sets to closed sets.

Let f(x,y) be a scalar function, and let F(x,y,z) be a vector field. Only one of...

Let f(x,y) be a scalar function, and let F(x,y,z) be a vector field. Only one of the following expressions is meaningful. Which one? a) grad f x div F b) div(curl(grad f)) c) div(div F) d) curl(div(grad f)) e) grad(curl F)

Let f: X→Y and g: Y→Z be both onto. Prove that g◦f is an onto function...

Let f: X→Y and g: Y→Z be both onto. Prove that g◦f is an onto function Let f: X→Y and g: Y→Z be both onto. Prove that f◦g is an onto function Let f: X→Y and g: Y→Z be both one to one. Prove that g◦f is an one to one function Let f: X→Y and g: Y→Z be both one to one. Prove that f◦g is an one to one function

Let F be a field. (a) Prove that the polynomials a(x, y) = x^2 − y^2,...

Let F be a field. (a) Prove that the polynomials a(x, y) = x^2 − y^2, b(x, y) = 2xy and c(x, y) = x^2 + y^2 in F[x, y] form a Pythagorean triple. That is, a^2 + b^2 = c^2. Use this fact to explain how to generate right triangles with integer side lengths. (b) Prove that the polynomials a(x,y) = x^2 − y^2, b(x,y) = 2xy − y^2 and c(x,y) = x^2 − xy + y2 in F[x,y]...

Prove Proposition 6.10 (Let f : X → Y and g : Y → Z be...

Prove Proposition 6.10 (Let f : X → Y and g : Y → Z be one to one and onto functions. Then g ◦ f : X → Z is one to one and onto; and (g ◦ f)−1 = f−1 ◦ g−1 ).

Let f: X→Y be a map with A1, A2⊂X and B1,B2⊂Y (A) Prove f(A1∪A2)=f(A1)∪f(A2). (B) Prove...

Let f: X→Y be a map with A1, A2⊂X and B1,B2⊂Y (A) Prove f(A1∪A2)=f(A1)∪f(A2). (B) Prove f(A1∩A2)⊂f(A1)∩f(A2). Give an example in which equality fails. (C) Prove f−1(B1∪B2)=f−1(B1)∪f−1(B2), where f−1(B)={x∈X: f(x)∈B}. (D) Prove f−1(B1∩B2)=f−1(B1)∩f−1(B2). (E) Prove f−1(Y∖B1)=X∖f−1(B1). (Abstract Algebra)

Let (X,dX),(Y,dY ) be metric spaces and f: X → Y be a continuous bijection. Prove...

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TOPOLOGY Let f : X → Y be a function. Prove that f is one-to-one and...

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