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).
 Justify all your work with good pictures, analytical work, explanations or a
combination.
 USE COMPLETE SENTENCES when explaining and writing your proofs so you
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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.
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Date_______________________________
Math 301 Summer 2019 Dr. Kimberly Vincent
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
a  b(modn)
and
c  d(modn)
then
ac  bd(modn).
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):
3 3
2
( 1)
1

4 5
1
3 4
1

 

 n
n
n n
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):
3 3
2
( 1)
1

4 5
1
3 4
1

 

 n
n
n n
. 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:
mN
, if
a  b(modn)
then
a b (modn)
m m
 .

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