Lesson 41.

NOTE: Be sure to read  Homework Format  and Homework Writing Policy BEFORE writing up your solutions to be turned in.

Due Friday, April 4

Sect. 5:

1. Give a simple geometric description of the set
     { (x, y) : there exist  r > 0  such that x² + y² = r² }.
   Prove your answer unless you are reasonably sure that every student in Math 323-1 would believe that a proof is not necessary because the answer is obvious.
   
   
2. Let  r  be a positive real number. A circle of radius  r  centered at the origin of a two-dimendional coordinate system is usually described by an equation  x² + y² = r².  Of course, an equation is NOT really a set. Give an accurate description/definition of a circle of radius  r  using set-builder notation, as follows:
   
   Let  r  be a positive real number. A circle of radius  r  centered at the origin is the set
   C_r = { ________________ : _______________________________________________________ }
   
   
3. (Cf. Lesson 37b.) Give a careful definition of a circle (in the plane).
   Begin by giving the definition, using set-builder notation, of a
    “circle with center  (a, b)  and radius  r ”,
   for given real numbers  a,  b,  and  r.
   Then define a circle in general as a set  C  for which there exists numbers  ...  such that  ... .
   
   
4. Give a careful definition of a nonvertical line (in the plane).
   Begin by giving the definition, using set-builder notation, of a
    “line with slope  m  and intercept  b ”,
   for given real numbers  m  and  b.
   Then define a line in general as a set  L  for which there exists numbers  ...  such that  ... .

NOTE: As stated on the Course Home Page, all due dates are tentative. Assignments, or parts of assignments, may be postponed to a later date.

Last modified Apr 3, 2008 10:31 AM

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