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gral coordinates and whose coordinates are relatively prime, i.e. of all of their common divisors, the greatest is 1. Prove this, i.e. prove: for any pair (x, y) = Qײ – {(0,0)}, there exists a related pair (z, w) (via R') such that (z, w) € ZX² - {(0,0)} and z and w are relatively prime.21

gral coordinates and whose coordinates are relatively prime, i.e. of all of their common divisors, the greatest is 1. Prove this, i.e. prove: for any pair (x, y) = Qײ – {(0,0)}, there exists a related pair (z, w) (via R') such that (z, w) € ZX² - {(0,0)} and z and w are relatively prime.21

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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The projective line and some geometry. With the set of rational numbers, Q (which we'll write using the familiar no- tation of fractions, e.g., knowing full well we may recall their definition as equivalence classes...), now in hand, we may define the (set of rational points of the) projective line, denoted P×¹(Q): Define a relation R' := {((x1, Y₁), (x2, Y2)) = (Qײ – {(0, 0)})^² |‡λ = Q-{0} such that (x1, Y1) = (\x2, \y2)} (25) ℃ (Qײ – {(0,0)})ײ. (26) We say pairs (x1, y₁) and (x2, y2) are related via R′ iff ((x1, y₁), (X2, Y2)) = R', i.e. there exists A € Q - {0} such that (x₁, y₁) = (\x2, \y2), and moreover, we say λ = Q = {0} "witnesses" the fact that (x₁, y₁) and (x2, Y₂) are related via R'. 17 35 = 12' After playing around with these pairs, e.g. (35, ) we might begin to believe there is always some related pair with integral coordinates, e.g. ((12.11). (12.11). 17) (1135, 12-17) (385,204) is related to (25, 17) with witness > Q- {0}. Better yet, it seems we can always such a related pair with inte- gral coordinates and whose coordinates are relatively prime, i.e. of all of their common divisors, the greatest is 1. Prove this, i.e. prove: for any pair (x, y) = Qײ – {(0,0)}, there exists a related pair (z,w) (via R') such that (z, w) € Zײ – {(0,0)} and z and w are relatively prime.21 Conversely, any pair (z, w) € Zײ – {(0, 0)} of relatively prime integers is already = 35 12' 12.11 € =
Transcribed Image Text:The projective line and some geometry. With the set of rational numbers, Q (which we'll write using the familiar no- tation of fractions, e.g., knowing full well we may recall their definition as equivalence classes...), now in hand, we may define the (set of rational points of the) projective line, denoted P×¹(Q): Define a relation R' := {((x1, Y₁), (x2, Y2)) = (Qײ – {(0, 0)})^² |‡λ = Q-{0} such that (x1, Y1) = (\x2, \y2)} (25) ℃ (Qײ – {(0,0)})ײ. (26) We say pairs (x1, y₁) and (x2, y2) are related via R′ iff ((x1, y₁), (X2, Y2)) = R', i.e. there exists A € Q - {0} such that (x₁, y₁) = (\x2, \y2), and moreover, we say λ = Q = {0} "witnesses" the fact that (x₁, y₁) and (x2, Y₂) are related via R'. 17 35 = 12' After playing around with these pairs, e.g. (35, ) we might begin to believe there is always some related pair with integral coordinates, e.g. ((12.11). (12.11). 17) (1135, 12-17) (385,204) is related to (25, 17) with witness > Q- {0}. Better yet, it seems we can always such a related pair with inte- gral coordinates and whose coordinates are relatively prime, i.e. of all of their common divisors, the greatest is 1. Prove this, i.e. prove: for any pair (x, y) = Qײ – {(0,0)}, there exists a related pair (z,w) (via R') such that (z, w) € Zײ – {(0,0)} and z and w are relatively prime.21 Conversely, any pair (z, w) € Zײ – {(0, 0)} of relatively prime integers is already = 35 12' 12.11 € =
in Qײ – {(0, 0)}. So, every equivalence class [x : y] of R' admits a representative (x, y) ≤ Zײ – {(0,0)} such that x and y are relatively prime. So, Prove there is a bijection between the set of equivalence classes of R' and the set of equivalence classes of R (as defined in "Relations toward functions"), so we had already constructed the (set of rational points of the) projective line en route to our original construction of the rational numbers! 21 Toward a proof of this, study the example provided; it seems we may simply clear denom- inators, and then factor out any residual common factors...
Transcribed Image Text:in Qײ – {(0, 0)}. So, every equivalence class [x : y] of R' admits a representative (x, y) ≤ Zײ – {(0,0)} such that x and y are relatively prime. So, Prove there is a bijection between the set of equivalence classes of R' and the set of equivalence classes of R (as defined in "Relations toward functions"), so we had already constructed the (set of rational points of the) projective line en route to our original construction of the rational numbers! 21 Toward a proof of this, study the example provided; it seems we may simply clear denom- inators, and then factor out any residual common factors...
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