Math 283 Spring 2013
Assignment 3 Solutions
1. Prove that A B C if and only if A B and B C .
Solution.
This is false. Let A = cfw_1, B = cfw_1 and C = cfw_2. Notice that everything holds, except
B C . Further, if A and B are as before, but C = cfw_1, 2, t
Math 283 Spring 2013
Assignment 4 Solutions
1.
P (A B ) = P (A) P (B )
Solution.
This is true.
()
()
Let X P (A B ). This means X A B and so X A and X B (since x X
means x A and x B ). Since X A, we have X P (A). Similarly, X B means
X P (B ). Thus, we h
Math 283 Spring 2013
Assignment 5 Solutions
n
1. For n N prove that
i(i + 1) =
i=1
n(n + 1)(n + 2)
.
3
Solution.
Notice that for n = 1, we have 1(1 + 1) = 2 =
n
If
i(i + 1) =
i=1
1(2)(3)
.
3
n(n + 1)(n + 2)
, adding (n + 1)(n + 2) to the left hand side gi
Math 283 Spring 2013
Assignment 6 Solutions
1. Prove that exponentiation to a positive odd power denes a strictly increasing function.
For n N, nd all solutions to xn = y n . (Hint: One possibility is to consider the cases
x < 0 < y , 0 < x < y and x < y
Math 283 Spring 2013
Assignment 7 Solutions
1. Prove that the natural numbers, the even natural numbers, and the odd natural numbers
form sets of the same cardinality.
Solution.
Every even natural number is obtained by doubling a unique natural number, so
MAT 283
EXAM #1 Solutions
2013-02-20
1. Consider A = cfw_1, 2, 3, 4, 5 , B = cfw_4, 5, 6 and C = cfw_1, 2 as subsets of [9] = cfw_1, 2, . . . , 9.
(a) (2 points) Is 2 A?
Yes
(b) (2 points) Is 2 B ?
No
(c) (2 points) Is 2 A?
No
(d) (2 points) Is cfw_2 A?
Y
MAT 283
Exam #1 Take-home Part
2012-02-27
Guidelines: You may use your textbook, your notes, or ask me, Ryan
Hansen, questions. You cant discuss the problems with anyone other than
myself.
Make sure to read, write and sign the last page.
Answer the questi
MAT 283
Exam #1 Take-home Solutions
6. Provide an argument or counterexample to support the truth value of the following
statements.
(a) (3 points) x R, x2 x = 0.
Solution: True since the value x = 0 or x = 1 satisfy the equation.
(b) (3 points) x R, x2 =
MAT 283
EXAM #2 Solutions
2013-03-20
1. Express each of the following sums using summation notation.
(a) (5 points) 4 + 5 + 6 + 7
7
Solution: One possibility is
i.
i=4
(b) (7 points)
1
111
+ + + +
246
20
10
Solution: One possibility is
i=1
1
.
2i
(c) (7 p
MAT 283
Exam #2 Take-home Part
2013-03-20
Guidelines: You may use your textbook, your notes, or ask me, Ryan Hansen,
questions. You cant discuss the problems with anyone other than myself.
Make sure to read, write and sign the last page.
Answer the questi
MAT 283
Exam #3 Take-home Part
2013-04-22
Guidelines: You may use books and notes. You cant discuss the problems with
anyone other than myself. Make sure to read, write and sign the last page.
Answer the questions in the spaces provided.
There are 3 quest
MAT 283
Exam #3 Take-home Part Solutions
2013-04-22
1. (10 points) Construct an explicit bijection from [0, 1] to (0, 1).
(Hint: One way to do it would be three parts. One for f (0), one for f
1
n
for n N and one
for everything else.)
Solution: Let f : [0
FINAL EXAM Solutions
2013-05-07
1. (7 points) Prove for sets A, B and C ,
A (B \ C ) (A B ) \ (A C ).
(Aside: In fact this is equal, not just a subset.)
Solution: Let (x, y ) A (B \ C ). This means x A and y B \ C , so y B and
y C . Since x A and y B , (x
MAT 283
Final Exam Take-home Part
2013-05-07
Guidelines: You may use books and notes. You cant discuss the problems with
anyone other than myself. Make sure to read, write and sign the last page.
Answer the questions in the spaces provided.
There are 5 qu
Math 283 Spring 2013
Presentation Problems 1 Solutions
1. If m is an even integer, then m + 1 is an odd integer.
Solution.
Let m be an even integer. This means that m = 2k for some integer k . Notice that
m + 1 = 2k + 1 and, since k was an integer, this m
Math 283 Spring 2013
Presentation Problems Solutions
1. An integer is even if and only if its square is even.
Solution.
We have two items to prove here. First, if an integer is even, its square is even. Let m
be an even integer. This means that m = 2k for
Name:
Math 283 Spring 2013
Assignment 2 Solutions
1. The statement below is not always true for real numbers x and y . Give an example where
it is false, and add a hypothesis (condition) on y that makes it a true statement.
If x and y are nonzero real num