Lesson 48.

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 followed.

Last modified Apr 23, 2008 10:11 PM

Postponed to Wednesday, April 23
(See email for details.)

April 23: The word “exactly” has been added in parentheses in a few places.
Sect. 8, 10:
Be sure to read the Induction Warnings and Induction Checklist before doing the induction part of these problems. Solutions which do not follow these guidelines will not be graded.
(There is a reason for the peculiar indentation format used for the problems below. You do not have to know the reason to solve these problems completely and correctly, but there is a reason.)

  1. (Use the ideas from Section 8 to give these definitions.) Let  S  be a set and  n  be a natural number.  Give the definition of each of the following:
     (*)  S  has 0 (zero) elements.  (“S  has 0 elements iff ... ” ?)
    (**) S  has (exactly)  n  elements.  ( ... )
    NEW(***) S  is a nonempty, finite set.
  1. Suppose  n  is a natural number and a set  S  has  n  elements.  Let  a  be anything. How many elements does  S \ {a}  have?  What about  S ∪ {a}?  Caution: There are, of course, two cases for each question.
    No explanation or proof necessary (but follow the usual rules for answering homework questions). You can use these facts where necessary in the following.
  1. Interlude:  Using the definition(s) in Problem 1, describe BRIEFLY what you have to prove in order to prove your answer to the first question in the preceding problem.*
  1. Back to the main train of thought:  Use induction to prove that if a set  S  of real numbers has (exactly)  n  elements (for a natural number  n),  then  S  has a least element.
    CAUTION. We have discussed frequently throughout the semester, and also in connection with induction, good and “bad” approaches to proving such statements. Try to avoid the bad. See also Notes, added April 20.
  1. Prove that every nonempty finite subset of the real numbers has a least element.
  2. Solve Exercise 10.30 for the case of (nonempty) finite subsets of the natural numbers.
* Regarding Problem 3:  You are NOT being asked to give the proof; you are NOT being asked HOW you would prove it; you are NOT being asked to start the proof. You are being asked to describe, based on the definition(s) you gave in the first problem, what you have to prove.

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


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 23, 2008 10:11 PM

Go to Lesson 48a.

Go to Lesson 49.

Back to Course Home Page