7. Consider the vector space F³ = = {(x, y, z)|x, y, z ≤ F}, where F is field. (a). let ~ be a relation on F³ \ {0, 0, 0}, defined by α = : (x₁, Y₁, 2₁) ~ ß = (X2, Y2, 72) if there exists some > € K s.t. λa = B. Show that X is an equivalence relation. (b). Describe the equivalent classes [a] in F3. (c). The projective plane over F, denoted by FP2 is the set of equivalent classes [a]. We interpret the primitive term as • point: [a], a € F³. ● line: {[a]|α = (x, y, z), ax+by+cz = 0} for some fixed a, b, c = F.
7. Consider the vector space F³ = = {(x, y, z)|x, y, z ≤ F}, where F is field. (a). let ~ be a relation on F³ \ {0, 0, 0}, defined by α = : (x₁, Y₁, 2₁) ~ ß = (X2, Y2, 72) if there exists some > € K s.t. λa = B. Show that X is an equivalence relation. (b). Describe the equivalent classes [a] in F3. (c). The projective plane over F, denoted by FP2 is the set of equivalent classes [a]. We interpret the primitive term as • point: [a], a € F³. ● line: {[a]|α = (x, y, z), ax+by+cz = 0} for some fixed a, b, c = F.
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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[Classical Geometry] How do you solve #7? The second picture is a hint (you don't need to solve the bullet points in the hint, just the asked question in the list of seven)
![1. Construct √an – bn using the ruler and compass, where a > b> 0 in an ordered field F
(assuming you have two points (0, 0) and (1,0), as always.)
2. Show that Q√3 = {a +b√3|a, b € Q} is a field by verifying the field axioms one by one.
3. Show that if (F, P) is an ordered field and a € F is such that a > 0, so is a ¹.
4. Let II be an ordered field.
(a). Explain what does 3 mean in F.
(b). Give an example of ordered field F where 3 does not have a square root in F
(c). Show that there is an equilateral triangle in II if and only if √3 € F.
5. Show the congruence axiom C3 for IIF (don't assume the field is Pythagorean)
6. Can we discuss incidence in IIF31? Can we discuss betweenness in IIF3₁1? Explain your
answer.
7. Consider the vector space F³ = {(x, y, z)|x, y, z € F}, where F is field.
(a). let ~ be a relation on F³ \ {0, 0, 0}, defined by
a = (x1, y₁, 21) ~ B = (x2, Y2, 22)
if there exists some > € K s.t. Aa= B. Show that A is an equivalence relation.
(b). Describe the equivalent classes [a] in F3.
(c). The projective plane over F, denoted by FP2 is the set of equivalent classes [a]. We
interpret the primitive term as
• point: [a], a € F³.
● line: {[a] a = (x, y, z), ax + by + cz = 0} for some fixed a, b, c = F.
(d). show that the line is well defined: if ax+by+cz = 0 holds for some a = = (x, y, z),
then it holds for any B = (x1, 9₁, 2₁) s.t. B € [a].
(e). How many points are there in F₂P²? List them.
(f). How many lines are there in F₂P2? List them.
(g). Show that F₂P2 is isomorphic to the Fano plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F73d0f296-b76f-4447-9ace-dbe53818487b%2F5633a4a2-719e-4958-ac4a-4a95ef5787fa%2F9ekr05q_processed.png&w=3840&q=75)
Transcribed Image Text:1. Construct √an – bn using the ruler and compass, where a > b> 0 in an ordered field F
(assuming you have two points (0, 0) and (1,0), as always.)
2. Show that Q√3 = {a +b√3|a, b € Q} is a field by verifying the field axioms one by one.
3. Show that if (F, P) is an ordered field and a € F is such that a > 0, so is a ¹.
4. Let II be an ordered field.
(a). Explain what does 3 mean in F.
(b). Give an example of ordered field F where 3 does not have a square root in F
(c). Show that there is an equilateral triangle in II if and only if √3 € F.
5. Show the congruence axiom C3 for IIF (don't assume the field is Pythagorean)
6. Can we discuss incidence in IIF31? Can we discuss betweenness in IIF3₁1? Explain your
answer.
7. Consider the vector space F³ = {(x, y, z)|x, y, z € F}, where F is field.
(a). let ~ be a relation on F³ \ {0, 0, 0}, defined by
a = (x1, y₁, 21) ~ B = (x2, Y2, 22)
if there exists some > € K s.t. Aa= B. Show that A is an equivalence relation.
(b). Describe the equivalent classes [a] in F3.
(c). The projective plane over F, denoted by FP2 is the set of equivalent classes [a]. We
interpret the primitive term as
• point: [a], a € F³.
● line: {[a] a = (x, y, z), ax + by + cz = 0} for some fixed a, b, c = F.
(d). show that the line is well defined: if ax+by+cz = 0 holds for some a = = (x, y, z),
then it holds for any B = (x1, 9₁, 2₁) s.t. B € [a].
(e). How many points are there in F₂P²? List them.
(f). How many lines are there in F₂P2? List them.
(g). Show that F₂P2 is isomorphic to the Fano plane.
Transcribed Image Text:Q7. First recall what is a equivalence relation.
Also note that the equivalence relation is put on the triple (x, y, z) = F³, not on F
e) f) are simple as long as you justified the equivalence relation well and checked that the line is indeed well-defined for the point in this plane(which is a collection of equivalence
classes)
g) can be justified by
• indicate rigorously what is a Fano plane. I hope you don't commit the mistake of the previous mistake again.
• Indicate rigorously what is F2P.
Now construct a map between them, and show that it is an isomorphism (recall what is an isomorphism between two models (of incidence geometry?))
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