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

Expert Solution

Colby Messinger answered 2 years ago

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 ).

5. Let X, Y and Z be sets. Let f : X ! Y and g...

5. Let X, Y and Z be sets. Let f : X ! Y and g : Y ! Z functions. (a) (3 Pts.) Show that if g f is an injective function, then f is an injective function. (b) (2 Pts.) Find examples of sets X, Y and Z and functions f : X ! Y and g : Y ! Z such that g f is injective but g is not injective. (c) (3 Pts.) Show that if...

The curried version of let f (x,y,z) = (x,(y,z)) is let f (x,(y,z)) = (x,(y,z)) Just...

The curried version of let f (x,y,z) = (x,(y,z)) is let f (x,(y,z)) = (x,(y,z)) Just f (because f is already curried) let f x y z = (x,(y,z)) let f x y z = x (y z)

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)

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)

3. Let F : X → Y and G: Y → Z be functions. i. If...

3. Let F : X → Y and G: Y → Z be functions. i. If G ◦ F is injective, then F is injective. ii. If G ◦ F is surjective, then G is surjective. iii. If G ◦ F is constant, then F is constant or G is constant. iv. If F is constant or G is constant, then G ◦ F is constant.

Let f : X → Y and g : Y → Z be functions. We can...

Let f : X → Y and g : Y → Z be functions. We can deﬁne the composition of f and g to be the function g◦ f : X → Z for which the image of each x ∈ X is g(f(x)). That is, plug x into f, then plug theresultinto g (justlikecompositioninalgebraandcalculus). (a) If f and g arebothinjective,must g◦ f beinjective? Explain. (b) If f and g arebothsurjective,must g◦ f besurjective? Explain. (c) Suppose g◦ f isinjective....

Let f: X-->Y and g: Y-->Z be arbitrary maps of sets (a) Show that if f...

Let f: X-->Y and g: Y-->Z be arbitrary maps of sets (a) Show that if f and g are injective then so is the composition g o f (b) Show that if f and g are surjective then so is the composition g o f (c) Show that if f and g are bijective then so is the composition g o f and (g o f)^-1 = g ^ -1 o f ^ -1 (d) Show that f: X-->Y is...

5) Let the function f : ℝ3 → ℝ3 be given by f(x, y, z) =...

5) Let the function f : ℝ3 → ℝ3 be given by f(x, y, z) = (2x + 2y, 2y + 2z, z + x). a) Prove that f is one to one and onto b) Find the inverse of f, i.e., f−1.

Consider the scalar functions f(x,y,z)g(x,y,z)=x^2+y^2+z^2, g(x,y,z)=xy+xz+yz, and=h(x,y,z)=√xyz Which of the three vector fields ∇f∇f, ∇g∇g and...

Consider the scalar functions f(x,y,z)g(x,y,z)=x^2+y^2+z^2, g(x,y,z)=xy+xz+yz, and=h(x,y,z)=√xyz Which of the three vector fields ∇f∇f, ∇g∇g and ∇h∇h are conservative?

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