Question

Let f:C⟶C be the function defined by f(z)=z⁵ for any z∈C.

Fill in the blanks in the blocks below, all labelled by capital-letter Roman numerals, with appropriate words so that they give respectively a proof for the statement (E) and a proof for the statement (F).
(a) Here we prove the statement (E) :
(E) The function f is surjective.
[We want to verify the statement ' ________(I)________ '.]_______(II)_______For this ζ, _____(III)_______ such that ζ=∣ζ∣(cos(θ)+isin(θ))._________(IV)_____________
By definition, z∈ C. Note that f(z)=z⁶ = _____(V)____ It follows that (VI)____________
(b) Here we prove the statement (F) :
(F) The function f is not injective.
[We want to verify the statement ' (I)_____________ ] (II)___________ Note that z₀,w₀ ∈C. Also note that (III)__________
We have (IV)__________ and (V) ______Then f(z₀ )=f(w₀ ). It follows that (VI)__________

Answer 1

To prove that the function f(z) = z^5 is surjective, we can use the polar form of a complex number and find a corresponding input for any arbitrary output. To prove that f is not injective, we can choose two different inputs that map to the same output.

Step-by-step explanation:

(a) To prove the statement (E), we want to show that the function f is surjective. Let's consider an arbitrary complex number ζ. We need to find a complex number z such that f(z) = ζ. Let's express ζ in the polar form as ζ = |ζ|(cos(θ) + isin(θ)). Now, we can choose z = |ζ|^(1/5)(cos(θ/5) + isin(θ/5)). Plugging this value of z into f(z), we have f(z) = z^5 = (|ζ|^(1/5)(cos(θ/5) + isin(θ/5)))^5 = |ζ|(cos(θ) + isin(θ)) = ζ. Therefore, f is surjective.

(b) To prove the statement (F), we want to show that the function f is not injective. Let's consider two different complex numbers z₀ and w₀. Without loss of generality, assume z₀ ≠ w₀. Now, let's evaluate f(z₀) and f(w₀). Since f(z) = z^5, we have f(z₀) = z₀^5 and f(w₀) = w₀^5. Since z₀ ≠ w₀, z₀^5 ≠ w₀^5. Thus, we have found different inputs z₀ and w₀ such that f(z₀) = f(w₀), which implies that f is not injective.