5. Let f: CC be the function defined by f(2)= 25 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). (The 'underline' for each blank bears no definite relation with the length of the answer for that blank.) (a) Here we prove the statement (E): (E) The function f is surjective. [We want to verify the statement (II) For this C (IV) By definition, z E C. Note that f(2)= 2³- It follows that (VI) (b) Here we prove the statement (F): (F) The function f is not injective. (III) [We want to verify the statement (II) Note that zo, wo E C. Also note that We have (IV) and It follows that (VI) (III) (V) (1) 9 such that (= (cos(0) + i sin(0)). (1) (V) 1 Then f(zo) = f(wo).
5. Let f: CC be the function defined by f(2)= 25 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). (The 'underline' for each blank bears no definite relation with the length of the answer for that blank.) (a) Here we prove the statement (E): (E) The function f is surjective. [We want to verify the statement (II) For this C (IV) By definition, z E C. Note that f(2)= 2³- It follows that (VI) (b) Here we prove the statement (F): (F) The function f is not injective. (III) [We want to verify the statement (II) Note that zo, wo E C. Also note that We have (IV) and It follows that (VI) (III) (V) (1) 9 such that (= (cos(0) + i sin(0)). (1) (V) 1 Then f(zo) = f(wo).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:5. Let f: CC be the function defined by f(z) = 25 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). (The 'underline' for each blank bears.
no definite relation with the length of the answer for that blank.)
(a) Here we prove the statement (E):
(E) The function f is surjective.
[We want to verify the statement
(II)
For this C,
(IV)
By definition, z E C. Note that f(z) = 2 =
It follows that
(VI)
(b) Here we prove the statement (F):
(F) The function f is not injective.
(III)
[We want to verify the statement
(II)
Note that zo, wo E C. Also note that
We have
(IV)
and
It follows that
(VI)
(III)
(V)
(1)
9
such that CC(cos(0) + sin(0)).
(1)
(V)
Then f(zo) f(wo).
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