Math 301 Summer 2019 Dr. Kimberly Vincent
Exam II-Direct Proof, Proof of Contrapositive, Proof by Cases, Proof by Contradiction,
Mathematical Induction
Write your solutions with only one question per page side of a sheet of paper. You may use both sides. (I will not read any work on this question sheet).
Follow all directions.
Justify all your work with good pictures, analytical work, explanations or a combination.
USE COMPLETE SENTENCES when explaining and writing your proofs so you complete your thoughts.
Sign and include the following statement in the pdf of your exam:
I acknowledge that all the work submitted for
the final exam in Math 351, spring 2020 is my
own original work.
I acknowledge that I did not share my work or
copy anyone else’s work as part of this exam.
I further acknowledge that if I shared my
answers or copied other’s work we both get
zeros on the exam.
Printed Name__________________________
Signature_____________________________
Date_______________________________
Math 301 Summer 2019 Dr. Kimberly Vincent
Ready? Time to begin…
1. (20 pts.) Write formal a proof for the following proposition: For all x and y in R, then |x+y|≤|x|+|y|.
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2. a. (20 pts) Prove the following: For all integers a and b and for all natural numbers n, if )(modnba and )(modndc then )(modnbdac .
b. (5 pts) write the converse of the proposition in part a.
c.. (15 pts) is the converse true? If yes, prove it. If not, provide a counter example.
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3. (20 pts.) Is the following statement true or false? If it is false provide a counter example. If it
is true provide an outline of a proof (you must give reasons for each claim). For each integer a,
if 3 does not divide a, then 3 divides 2a2+a.
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4. (20 pts.) Write a formal proof for the following theorem. Theorem: If x+y if irrational, then x
is irrational or y is irrational.
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5. (20 pts.) Write a formal proof for the following:
(a-3)b2 is even if and only if a is odd or b is even.
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6. (5 pts.) a. What is the first n that P(n) is true? P(n): 33
2
)1(
1 …
54
1
43
1
n
n
nn
b. (20 pts.) Use mathematics induction to prove (write a formal proof). For all n ϵ N, where n is greater than or equal to ? (the answer form part a) P(n) is true,
where
P(n): 33
2
)1(
1 …
54
1
43
1
n
n
nn . Be sure to state which of the three types of
mathematical induction you are using.
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Bonus: You will not lose points if you do this wrong. But you can earn extra points for correct
response(s).
Let n and m be natural numbers. Let a and b be integers. Prove the following:
Nm , if )(modnba then )(modnba mm .
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