Let E denote the set of all ordered pairs (x1, x2) of real numbers. Consider the

Question

Let E denote the set of all ordered pairs (x1, x2) of real numbers. Consider the following interpretation of the primitive concepts of Incidence Geometry...

Directions: Let E denote the set of all ordered pairs (x, x2) of real numbers. Consider the following interpretation of the primitive concepts of Incidence Geometry point P element P (x1,x2) of E line 1 subset 1 C E of all solutions (ri,T2) to an equation of the form aixut a2T2 c where al, a-, c are real numbers and (a1, a2)(0,0). P11 PEI. The goal of this series of problems is to use your knowledge of algebra to verify that this interpretation is a model of the axioms of incidence having the the Euclidean Parallel Property 1. To verify 1-3, show that if the points (0, 0), (1,0), and (0,1) in E satisfy an equation of the form aizi +a222 = c then (ai, a2) = (0,0), thus they cannot belong to a line in this interpretation. 2. To verify I-2, you need to start with equation ai21 +a2z2-c defining a line and find formulas for two specific points P-(zi, za) and Q (yi, p) in terms of ai, a2, and c which solve this equation. Hint: Consider tuo cases: Case 1: a20, and Case 2: a2 = 0 (so that al 0, otherwise (a1, a2)-(0,0)).

Explanation / Answer

1. Solution: Points (0,0), (1,0),   and   (0,1) in E.

We have equation a1x1 + a2x2 = c

For the points (0,0)   

a1(0) + a2(0) = c or   c = 0

Now

For the points (1,0)   

a1(1) + a2(0) = c ,

since c = 0 , thus a1 = 0.

and

For the points (0,1)   

a1(0) + a2(1) = c ,

or   a2 = 0

Therefore (a1, a2 ) = (0,0)

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.