Math 2710 (Roby)
Practice Final Solutions
Spring 2013
This is a closed book, closed note exam, except that you may have one 4 6 inch notecard
with anything you like written on it front and back. You may use a calculator. Please do
not discuss this exam wi
1/19/17
Exercise 5.) Satisfying definition 4, a=2n and b=2m. a+b have to = an even
number so substituting a and b into the equation, 2n+2m has to equal an
even number. Factoring out a 2, you get 2(n+m) so it proves that 2(any
number) = an even number. For
3/23/17
Fact: Let b, m with m > 0. If we divide b by m, the remainder is the integer
satisfying
0r<m
b r (mod m)
Exercise 98.) Remainder in 427 15? (425=b, 15=m)
Notice 42 16 1 (mod 15)
Using exercise 93(b), 427 = 42425 = 1425 (mod 15)
427 = 4264 = (42)
2/21/17
Exercise 61.) b.) We will prove B A. Let b B so b= 15n-24 for some n . We
have b= 15n-30+6 = 15(n-2)+6 A = cfw_15n+6 | n so B A.
Exercise 62.) d.) A=C, A B, B A is false
Exercise 63.) Let a,b and let A= cfw_ma+nb | m,n
Let B= cfw_ma+nb | a,b,m,n
3/7/17
H.W. #19.) If a , then 2 a = a or 2a
Prove 2 cases: a is odd: we get 2a (ex.70)
a is even: we get a
A, B sets
A or B is not equivalent to A B
x is an element of A or B is equivalent to x A B
Distinguish between elements and sets
In ex. 19, a 2 a
2/14/17
Exercise 44.) The remainders in the Euclidean form a strictly decreasing sequence
of non-negative integers r1 > r2 > r3 > and such a sequence cant continue infinitely
(infinite descent)
Lemma 45.1: Let a,b and write b=qa+r for some q,r with 0 r <
4/4/17
Exercise 115.) b.) Let a and b be irrational.
Assume ab is rational, a 0.
ab = c/d c,d
a = x/y x,y
[(y/x)x/y] b = c/d(y/x)
b= cy/dx b is rational which is a contradiction so ab is irrational
Exercise 116.) a.) If a,b are both irrational, then a+b
MATH 2710, SPRING 2014
HOMEWORK #1SOLUTIONS
JOHANNA FRANKLIN
This assignment will be due on Thursday, January 30 at the beginning of class. Remember to
show your reasoning and name the classmates you worked with. Your solutions should be detailed
enough t
MATH 2710, SPRING 2014
HOMEWORK #4SOLUTIONS
JOHANNA FRANKLIN
This assignment will be due on Thursday, February 27 at the beginning of class. Remember to
show your reasoning and name the classmates you worked with. Your solutions should be detailed
enough
MATH 2710, SPRING 2014
HOMEWORK #3SOLUTIONS
JOHANNA FRANKLIN
This assignment will be due on Thursday, February 13 at the beginning of class. Remember to
show your reasoning and name the classmates you worked with. Your solutions should be detailed
enough
MATH 2710, SPRING 2014
HOMEWORK #2SOLUTIONS
JOHANNA FRANKLIN
This assignment will be due on Thursday, February 6 at the beginning of class. Remember to
show your reasoning and name the classmates you worked with. Your solutions should be detailed
enough t
MATH 2710, SPRING 2014
HOMEWORK #5SOLUTIONS
JOHANNA FRANKLIN
This assignment will be due on Thursday, March 6 at the beginning of class. Remember to
show your reasoning and name the classmates you worked with. Your solutions should be detailed
enough that
Name:
Midterm 1
Math 2710 Spring 2014
Professor Hohn
Instructions: Turn o and put away your cell phone. No calculators or electronic devices are
allowed. Show all of your work! No credit will be given for unsupported answers or illegible
solutions.
Questi
Midterm 1 - Preview
Math 2710 Spring 2014
Professor Hohn
Instructions: These exercises are to be worked on alone! You may use your notes and the textbook, but you are not allowed to ask for, receive, nor give others assistance on these
exercises.
1. Let a
Math 2710 (Roby)
Practice Midterm #2
Spring 2013
1. Give a careful proof by induction that for every positive integer n
12 + 32 + 52 + + (2n 1)2 =
n(2n 1)(2n + 1)
3
For the base case n = 1 we get LHS=12 = 1 and RHS=
holds.
1(1)(3)
= 1, so the equation
3
N
Math 2710 (Roby)
Practice Midterm #1
Spring 2013
1. Use a truth table to check whether the statement (P AND Q) = R is equivalent to
the statement P = (Q = R).
A complete truth table looks as follows.
P
T
T
T
T
F
F
F
F
Q
T
T
F
F
T
T
F
F
R P AND Q Q = R (P
2/28/17
H.W. #4.) #13.) Prove gcd(a,a+1)=1 for all a .
One way: Use Euclidean Algorithm:
a+1= 1a + 1
a= 1a + 0
The division algorithm: b=qa + r; 0 r < |a|
We need to check special cases where 0 r < |a| is not true.
Exercise 79.) We will prove that 3n2+5n+