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(a) Prove that the function f is well-defined; that is, the value f(q) of the function at a rational number q € Q does not depend on the representation of q as a fraction. (b) Prove that the function f is injective. (c) Prove or disprove that the function f is surjective.

(a) Prove that the function f is well-defined; that is, the value f(q) of the function at a rational number q € Q does not depend on the representation of q as a fraction. (b) Prove that the function f is injective. (c) Prove or disprove that the function f is surjective.

Advanced Engineering Mathematics
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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Prove the following step by step in detail, please elaborate as much as possible and explain the steps, Im very lost on how to even approach this

For any two real numbers a and b, define the subset S(a,b) of R XR as S(a,b) = {(x, y) = R×R: ax+by=0}. You may use (without providing a proof) the fact that S(a,b) = S(c,d) if and only if ad = bc. Next, consider the collection C = {S (a,b) a,b ≤ R}, of sets and define the function f: Q→ C by ƒ (7) = S(a,b); for any Q. (a) Prove that the function f is well-defined; that is, the value f(q) of the function at a rational number q EQ does not depend on the representation of q as a fraction. (b) Prove that the function f is injective. (c) Prove or disprove that the function f is surjective.
Transcribed Image Text:For any two real numbers a and b, define the subset S(a,b) of R XR as S(a,b) = {(x, y) = R×R: ax+by=0}. You may use (without providing a proof) the fact that S(a,b) = S(c,d) if and only if ad = bc. Next, consider the collection C = {S (a,b) a,b ≤ R}, of sets and define the function f: Q→ C by ƒ (7) = S(a,b); for any Q. (a) Prove that the function f is well-defined; that is, the value f(q) of the function at a rational number q EQ does not depend on the representation of q as a fraction. (b) Prove that the function f is injective. (c) Prove or disprove that the function f is surjective.
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