Math 225 004 One-to-One Correspondences Worksheet
1. Find a one-to-one correspondence between each of the following pairs of sets.
(a) cfw_a, b, c, d, e, f , cfw_2, 4, 8, 16, 32, 64
f = cfw_(a, 2), (b
Math 225 004 Indirect Proofs
1. Write a proof by contraposition to show that for any real number x, if x2 + 2x < 0,
then x < 0.
Proof. Assume that x is a non-negative real number. Then 2x is also non-
Additional problems based on sample space & counting
Rules:
1. A coin is flipped 10 times and the sequence of heads and tails is observed. The number of
possible 10-tuplets that result in four heads a
MA 225 Exam 1-A
Name:
Problems Answer problems 1-5 on your own paper, and problems 6-8 on this sheet.
1. (10) Suppose we know that P Q and ( P ) Q. What can we conclude?
2. (10) If we know that R Q an
MA 225 Exam 1
Name:
1. (12) Every rational between 0 and 1 has an Egyptian fraction representation.
(a) Which of the following is/are equivalent to the above statement? Place a next to those
that are
MA 225 Exam 1-C
Name:
Problems Answer problems 1 and 2 on this sheet, and problems 3-6 on separate paper.
1. (10) Which of the following are eligible for a proof by induction? Put a 2 next to the
ones
Prove x1 x1+x=12using the formal definition of limit of a function.
Proof: Assume the x1x1+x=12. Let >0 arbitrary. Choose =min13, 6.
Let x dom(f) s.t. 0 < |x-1| < .
Then |f(x) - (12)| = |x1+x-12| = |x
For the following sequences determine whether they are convergent or divergent. For the
convergent one(s) make an educated guess what the value of the limit is and prove your claim
using the formal de
Question 2: Consider the conjecture given below. If you believe the conjecture is true, give a
formal mathematical proof. If you believe the conjecture is false, give a specific counterexample
and pro
Question 1: Suppose a, b and c are integers. What can we say about whether ab+ac is always
even or always odd. If you believe that ab+ac must always be even, then write a conjecture to
this effect and
MA225: Foundations of Advanced Mathematics
Test 1 : Solutions
(1) (10 pts.) Given the statement If G is planar then G is 4-colorable
and does not have a K5 minor.
(a) Write the converse.
If G is 4-co
MA225: Foundations of Advanced
Mathematics:
HW 9 Solutions
1. 5.1
(4) Show that if h : A C and g : B D are one-to-one correspondences, then f : A B C D given by f (a, b) = (h(a), h(b) is a
one-to-one
MA225: Foundations of Advanced Mathematics
Homework 1 Solutions
Section 1.1
10b)
A B B AB
T F T
T
Therefore, A B is a tautology.
10b)
A B A A B ( A B)
T F F
F
T
Therefore, ( A B) is a tautology.
13a)
MA225: Foundations of Advanced
Mathematics:
HW 7 Solutions
1. Section 3.2
(1f) = on N
Not reexive, since n = n for any n N.
Symmetric, since x = y y = x.
Not transitive, since 1 = 2 and 2 = 1 but 1
MA225: Foundations of Advanced
Mathematics:
HW 6 Solutions
As I mentioned in class, rewrites for HW 6 are due Monday, March
16th, after spring break. I have posted the solutions for the graded
problem
MA225: Foundations of Advanced Mathematics
Homework 5 : Solutions
These are the solutions to the ungraded problems which are not in contained in the back of the book.
1. Section 2.1
(5b) True.
(5d) Tr
MA225: Foundations of Advanced Mathematics
Homework 2 : Solutions to ungraded problems
Suppose a, b, c and d are integers.
(2) Prove that if ab|c then a|c.
If a, b, and c are integers and ab|c then a|
MA225: Foundations of Advanced Mathematics
Homework 4 : Solutions
1. Section 1.5
(3h) Let x, y, andz be integers. If x does not divide yz then x does not
divide z.
Proof. We will use the contrapositiv
MA225: Foundations of Advanced Mathematics
Homework 3 : Solutions to ungraded problems
1. Section 1.4
(7d) If a is an integer then a(a + 1) is even.
Proof. Let a be an integer.
Case 1: a is even.
Sinc
MA225: Foundations of Advanced Mathematics
Test 3 : April 10, 2015
(1) (15 points) State whether each of the following propositions is True or
False. Justify your answers either by giving a proof or a
MA225: Foundations of Advanced Mathematics
Test 2 : March 6, 2015
(1) (12 pts.) Let A = cfw_a, . . . , z and B = cfw_1, . . . , 10. Then
R = cfw_(m, 8), (p, 1), (r, 2), (m, 10), (f, 3) is a relation f
MA 225 004 Homework 7 Solutions
Exercises 5.1:
1. Proof. (R) For any set A, f : A A such that f (x) = x is 1-1 and onto. So A A.
(S) Suppose that A B for sets A, B. Then there is a one-to-one correspo
MA 225 004 Homework 5 Solutions
Exercises 4.1:
2. a) r = cfw_(5, 3), (5, 4); b) r = cfw_(5, 3), (6, 4); c) cfw_(5, 3), (6, 4), (7, 5); d) cfw_(5, 3), (6, 4), (7, 4)
6. a) (1, 1) are both in the relati
MA 225 Worksheet: Quantiers
The point of quantiers is not just to scatter some s and s and call it a day! Its very
important to say what you mean. One special avor of saying what you mean is not sayin
MA 225 Problem Set 6
Reading Read sections 3.1 and 3.2 of the text.
Exercises First do these problems without consulting the textbook or your classmates.
1. Exercises 3.1.1 and 3.1.4.
2. Sketch the re
MA 225 Problem Set 1
Reading Read the Preface to the Student in the textbook.
Exercises First do these problems without consulting the textbook or your classmates.
1. Suppose that P , Q, and R are pro