Lesson 54a.
Due Friday, May 2
- Sect. 12 (and clover smelling):
-
- Review the result of Lesson 52 and then solve
Exercise 12.12(a).* Be sure to read the footnote below.
- Both this Lesson and Lesson 54 asked you to “review the result
of Lesson 52”. Explain the relevance of this.
(And if you ignored Lesson 52 before, ask yourself why you did so.)
- Refer to Theorem 12.7. In this theorem, a set
C is defined in terms of subsets A
and B of the real numbers. Denote the
set C defined in this way as
A + B. Determine this
“sum” when
A = {-1, 1} and
B = {1, -1}. Be sure to use the definition
carefully as it is given. Errors here will be considered very serious.
- Suppose it were true, in Exercise 12.12, that
(f + g)(D) =
f(D) + g(D) (using the notation
of the preceding problem). Explain why it would follow that equality
holds (instead of ≤ ) in
12.12(a). (Formal proof not required; explain why it follows from
previous ideas and noted theorems.)
- Give an example of a set
D of real numbers containing exactly two elements, and
functions f and g with domain
D, for which (f + g)(D)
≠ f(D) + g(D).
- Solve Exercise 12.12(b).
* When solving Exercise 12.12(a), don't fill your proof with lots of
unnecessary “for all”. You know how to prove universally
quantified statements, and you don't do it by repeating “for all”
throughout the proof. You can use “for all” at the
beginning, if useful, to summarize your assumptions, and you can
use “for all” at the end, if useful, to summarize your
results, but don't repeat it throughout the proof. This guideline, of
course, does not apply only to Ex. 12.12(a).
- Prove the middle inequality in Exercise 12.8. As part of your
proof, be sure to justify the existence of the inf and
the sup.
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.
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 29, 2008 8:09 AM
Go to Lesson 48a (due April 28).
Go to Lesson 55 (due April 28).
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