Solutions to Math 305 Midterm Exam #1
1. (a) If S R is a set, then a is the supremum of S if the following two conditions hold:
(i) a s for every s S , and (ii) if a s for all s S , then a a . The real number b is
the inmum of S if the following two condi
Math 305 Fall 2011
Open and Closed Sets
Denition. Suppose that S R is a set. We say that a point x S is an interior point
of S if there exists some > 0 such that N (x; ) S. We write int S to denote the set of
all interior points of S. We say that a point
Mathematics 305 Fall 2011 Final Exam Solutions
1. Suppose that E R is nonempty and let x R.
(a) We say x int E if there exists some > 0 such that N (x; ) E c = .
(b) We say x bd E if N (x; ) E = and N (x; ) E c = for every > 0.
(c) We say x cl E if N (x;
Math 305 Fall 2011
Assignment #3
This assignment is due at the beginning of class on Tuesday, September 27, 2011.
1.
Exercises #12.3 and #12.4 on page 127.
2.
Exercise #12.5 on page 127.
3.
Exercise #12.9 on page 127.
Math 305 Fall 2011
Assignment #2
This assignment is due at the beginning of class on Tuesday, September 20, 2011.
1.
Read Sections 1 and 2 on pages 117 as a reminder of some basics of logic. This material
will appear throughout the course as we prove math
Math 305 Fall 2011
Assignment #1
This assignment is due at the beginning of class on Thursday, September 15, 2011.
1.
Read Section 5 on pages 3645 and do practice problems 5.2, 5.6, 5.9 ,5.11, 5.12, 5.14, 5.15,
and 5.18. Solutions may be found on pages 45
Math 305 Final Exam Thursday, December 15, 2011
This exam is worth 100 points.
This exam has 9 problems and 2 numbered page.
You have 3 hours to complete this exam. Please read all instructions carefully, and check
your answers. Show all work neatly and i
Math 305 Fall 2011
Solutions to Assignment #1
2.
To prove that (Ac )c = A, we need to verify the two containments (Ac )c A and A (Ac )c .
We will begin by showing that (Ac )c A. Suppose that x (Ac )c . By denition of complement,
this means that x (Ac ). B
Math 305 Fall 2011
Solutions to Assignment #3
Exercises 12.3 and 12.4. Suppose that S is the subset of R given in each problem.
(a) sup S = 3, max S = 3, inf S = 1, min S = 1
(b) sup S = , max S = , inf S = 3, min S = 3
(c) sup S = 4, max S = 4, inf S = 0
Math 305 Midterm Exam #2 November 15, 2011
This exam is worth 60 points.
This exam has 6 problems and 1 numbered page.
You have 75 minutes to complete this exam. Please read all instructions carefully, and check
your answers. Show all work neatly and in o
Math 305 Midterm Exam #1 October 13, 2011
This exam is worth 60 points.
This exam has 6 problems and 1 numbered page.
You have 75 minutes to complete this exam. Please read all instructions carefully, and check
your answers. Show all work neatly and in or
Math 305 Fall 2011
The Density of Q in R
The following two theorems tell us what happens when we add and multiply by rational
numbers. For the rst one, we see that if we add or multiply two rational numbers together,
then the result is necessarily a ratio