Math 3283W
10/13/2016
Functions
1. Suppose f : A B and S and T are subsets of A. Prove or give a counterexample.
(a) If S T then f (S) f (T ).
Proof. Suppose S T . Let x f (S). Then there exists y S such that f (y) = x. Since S T ,
y T . Thus x = f (y) f
Homework #8
Math 3283W  Fall 2016
The following problems will be relevant for your writing quiz on Thursday, 11/3/16.
Skill / Computational Problems. These problems are not handed in or graded, and do not involve
enough writing to be the basis of a writi
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Problem 1 (Lay 3.4.1). Let S R. Mark each statement True or False.
Justify each answer.
Solution. (a) int S bd S = . True. Suppose x int S. Then there
exists > 0 such that N (x, ) S, so x cannot be in the boundary
of S. On the othe
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Homework #2
Math 3283W  Fall 2016
The following problems will be relevant for your writing quiz on Thursday, 9/22/16.
Skill / Computational Problems. These problems are not handed in or graded, and do not involve
enough writing to be the basis of a writi
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Math 3283W  Fall 2016
Exam 2
12/8/2016
Time Limit: 50 Minutes
Name (Print):
Solutions
Your TAs Name:
This exam contains 5 pages (including this cover page) and 5 problems. Check to see if any pages are
missing. Enter all requested information on the top
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Math 3283W
12/6/2016
Exam 3 Review Partial Answers
1
Sequences
Note: None of these answers include complete proofs! They would not receive full credit on
the exam!
1. From the definition of convergence, show that limn
sin n
n
= 0.
Remember that 1 sin(x) 1
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Homework #7
Math 3283W  Fall 2016
The following problems will be relevant for your writing quiz on Thursday, 11/3/16.
Skill / Computational Problems. These problems are not handed in or graded, and do not involve
enough writing to be the basis of a writi
third
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Math 3283W  Fall 2016
Exam 1
10/6/2016
Time Limit: 50 Minutes
Name (Print):
Your TAs Name:
This exam contains 5 pages (including this cover page) and 6 problems. Check to see if any pages are
missing. Enter all requested information on the top of this pa
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Math 3283W Fall 2016  Writing Quiz #5
Thursday 10/20/16  20 minutes
Name:
Remember: your work on in the writing portion of this quiz will be graded on the quality of your writing and
explanation as well as the validity of the mathematics. (5 Points)
Def
EXAM 1 REVIEW SOLUTIONS
Problem 1. Prove De Morgans laws:
(A B)c = Ac B c
(A B)c = Ac B c .
Proof. The main idea of these proofs is that negation changes and into
or, and vice versa. Therefore, we only prove the first law:
() Suppose x (A B)c . This means
Math 3283W
11/1/2016
Interiors & Boundaries
1. For each set S below find S and int S.
(a) S = [0, 1] : S = cfw_0, 1, int S = (0, 1)
(b) S = (0, 1) : S = cfw_0, 1, int S = (0, 1)
(c) S = [0, 1) : S = cfw_0, 1, int S = (0, 1)
(d) S = Q : S = R, int S =
(e)
Math 3283W
9/15/16
Logic and Negation Practice
1. Negate the following statements.
(a) Some pencils are red.
All pencils are not red.
(b) All chairs have four legs.
Some chairs have three legs
Some chairs have less than four legs
Some chairs have five leg