Lesson 41c.

NOTE: Be sure to read  Homework Format  and Homework Writing Policy BEFORE writing up your solutions to be turned in.
PROBLEMS WILL NOT BE GRADED AND NO CREDIT WILL BE GIVEN if these guidelines are not follows.

Due Monday, April 21

Sect. 5:

Do problem 2c below. Follow the instructions carefully.

1. [Not assigned.]
2. 1. [Not assigned.] Let  r  be a positive real number. A circle of radius  r  centered at the origin in a two-dimensional 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 = { ________________ : _______________________________________________________ }
      
      
   2. [Not assigned.] Define a circle centered at the origin in general as a set  C  for which there exists a number  ...  such that  ... .
      (This is part (b) of Problem 2; you have just done part (a).)
      
      In other words  (Hold My Hand YouTube Video),  fill in the blanks in the following definition (all you need to do is to copy this definition onto your paper with the blanks filled in correctly; no more writing is required for this problem):
      A circle centered at the origin is a set  C  for which there exists a number _______ such that  C  [is or =] _________ .
      
      Also fill in the blanks in the following variation of this:
      A set  C  is a circle centered at the origin iff there exists a number _______ such that __________________ .
      
      
   3. Refer to the two forms of the definition requested in part (b) just above. Here is a correct version of the first definition requested:

      Defn 1   A circle centered at the origin is a set  C  for which there exists a number  r > 0  such that
      C = { (x, y) : x² + y² = r² }.

      Consider the following alleged solution of the second version requested:

      Defn 2   A set  C  is a circle centered at the origin iff there exists a number  r > 0  such that for all  (x, y)  in  C,  x² + y² = r² .

      1. Consider the empty set. Is it a circle according to Defn 1? According to Defn 2? Explain (of course).
         
         
      2. Find an example of a set containing exactly one point which is NOT a circle according to Defn 1, but is a “circle” according to Defn 2. Prove your answer.
         
         
      3. Find and sketch an example of a set containing uncountably many points (i.e., an uncountable set) which is NOT a circle according to Defn 1, and which no reasonable person would call a circle, but which is a “circle” according to Defn 2. To be specific, choose a connected set (in the intuitive sense of “connected”) consisting entirely of points which are 2 units from the origin.
         
         Prove your answer — i.e., prove that Defn 1 is not satisfied but Defn 2 is satisfied (in order to do this, you should give a set-builder description of your example). You do not have to prove that your example is uncountable or connected.

NOTE: Be sure to read  Homework Format  and Homework Writing Policy BEFORE writing up your solutions to be turned in.
PROBLEMS WILL NOT BE GRADED AND NO CREDIT WILL BE GIVEN if these guidelines are not follows.

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 16, 2008 9:45 PM

Go to Lesson 42 (due Apr 9).

Go to Lesson 49.

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