Math 235 - Dr. Miller - HW #3: Proof by Contrapositive - SOLUTIONS, 2/14/14
Please work each proof on a separate page so that I can return them 1-2 at a time if necessary.
Write your name on each page.
Problem #1 is due by Monday, 2/10/14, for early feedb
Math 235 - Dr. Miller - Topics List for Final Exam - Wednesday, May 9, 10:30am-12:30pm
Carefully review the items below with reference to your notes, text, activities/handouts, homework, and
1. Write or rewrite mathematically
Math 235 - Dr. Miller - Exam #3 - SOLUTIONS
1. [18 pts - 6 each] For each statement below, rewrite it in standard if-then form, write
what you would assume in the type of proof specied, what you need to show (labeled
NTS and in parentheses) in that type o
Math 235 - Dr. Miller - Exam #2 - 3/30/12 - SOLUTIONS
1. [10 pts] Prove carefully and rigorously via an appropriate denition: Let x be a real
number. If x is rational, then x + 2 is rational. (You must use a denition; you may
NOT simply invoke the homew
Math 235 - Dr. Miller - Topics List for Exam #1, Spring 2014 - Wednesday, Feb. 19, 2014
Study this list with reference to your notes, text, and graded and ungraded HW. Strive to master the concepts, explanations, and proof techniques in general; just memo
Math 235 - Dr. Miller - Bonus Problem for Exam #1, Spring 2014 - SOLUTIONS, 02/24/14
This problem is worth an additional 10 points added to your Exam #1 score. It is individualized so that only genuine collaboration can occur.
Assign numeric values to let
Math 235 - Dr. Miller - HW #1: Logic Review - SOLUTIONS, 1/29/14
1. Negate each statement below.
(a) x is composite and y is negative.
x is not composite (x is prime or a unit) or y 0.
(b) sin A > 0 or A is below the x-axis.
sin A 0 and A is on or above t
Math 235 - Dr. Miller - Topics List for Exam #2 - Spring 2014
1. Set up what you assume and need to show for a given or conclusion statement.
2. Recognize what kinds of statements are LEGITIMATELY proved by example and
what kind can be di
Math 235 - Dr. Miller - HW #2, Part B: Direct Proof - SOLUTIONS, 2/5/14
You have already turned in Part A, which was to prove Problems #5g and #7k from the
problem set on page 37 of our textbook. Please write each proof below on a SEPARATE
sheet of paper,
Math 235 - Dr. Miller - Exam #1, Spring 2014 - SOLUTIONS, 02/19/14
There are THREE 20-point proofs on the exam. Budget your time carefully.
1. [28 pts - 7 each] Negate each statement below, writing your answer in clear, correct
mathematical language, and
Math 235 - Dr. Miller - HW #6: General Set Proofs - SOLUTIONS, 03/14/14
1. (a) Refer to the books proof of Theorem 2.2.1(m). Rewrite that proof, annotating the end of each
line with the word defn or logic to indicate which was used.
Theorem 2.2.1(m). For
Math 235 - Dr. Miller - HW #5: Subset and Set Equality Proofs - SOLUTIONS, 03/07/14
1. (a) Prove that T = cfw_8n 7 | n Z is not a subset of 3Z.
Proof. (NTS: there exists x T where x 3Z.) Consider 17 = 8(3) 7 (where
3 Z). 17 T (because it has the form 8 ti
Math 235 - Dr. Miller - HW #4AB: Basic Proof Wrap-Up and Examples in Proofs - SOLUTIONS, 2/28/14
1. Prove that the graphs of y = x2 + 4x + 3 and y + 1 + (x 2)2 = 0 cannot intersect.
Proof. Suppose that these graphs intersect. (NTS: any contradiction) At t
Math 235 - Dr. Miller - Exam #1 - Feb. 24, 2012 - SOLUTIONS
This exam is worth 100 points; save plenty of time for the proofs at the end.
1. [18 pts - 6 each] Identify each sentence below as a true statement, a false statement, or
not a statement, informa