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.

Expert Solution

Colby Messinger answered 2 years ago

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....

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)

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?

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

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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...

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

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

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