Intro to Higher Math
Practice Midterm I
Instructor: David Hansen
Please explain your reasoning and prove your claims. Each problem is worth 20 points.
1. Let X be a finite set with n elements, and let Y be a finite set with 3 elements. How
many functions

V2000: Review for Midterm 1, Feb 18, 2016
This exam will have 4 questions each worth 15 points. All answers should be clearly
written with all statements justified (but not necessarily formally proved you have
to use your judgement about that.) NO CALCULA

1. Notes on Dedekind cuts
Definition 1.1. A subset L Q of the rationals is called a Dedekind cut if
(I) L is proper (i.e. L 6= , L 6= Q);
(II) L has no maximal element;
(III) for all elements a, b Q with a < b, b L = a L.
Example 1.2. (i) If a Q, the open

V2000: Notes on sequences: cf. DM p 148, Daepp-Gorkin Ch 19, 20.
Defn: Let (an )n1 be a sequence and L R. Then we say lim an = L if for all > 0 there
n
is N such that
|an L| < for all n N.
This is equivalent to the condition
> 0, N,
n N = |an L| < .
If a

V2000: Review for Midterm 1, Feb 18, 2016
By the way: the midterm is CLOSED BOOK no notes etc. are allowed.
Answers to the questions
Ex 1. (i) Denote by [a]n the set of all integers congruent to a modulo n. Show that
the operation
[a]n + [b]n = [a + b]n
i

V2000: Midterm 1, 10:10-11:25 am February 18, 2016
Here are comments on the exam which should help with rewriting.
Question 1: Let X, Y be sets and f : X Y any function.
(i) (2 pts) For A X and C Y define the sets f (A), f 1 (C).
I want a precise definiti

MATH 2000 Spring 2016, HW 11; due April 21 in class
Ex 1: Prove that the relation 4 on sets is reflexive and transitive. Is it antisymmetric?
Ex 2: Let X, Y be finite sets with |X| = m, |Y | = n. Prove the following statement by
induction on m 1: If there

MATH V2000, Spring 16. Review for Midterm II.
The exam covers Ch 3.5, Ch 4.1-4.2, Ch 7.1-7.3, Ch 5.1-2 and parts of 5.3 (up to the
end of Example 5.22 on sequences). Also we did rather little on continuity and I will NOT
ask you explicitly about results f

Homework 12: due Thursday April 28, 2016
Exercise 1: The construction of the rational numbers from the integers. Note that although you
cannot divide in Z you can cancel nonzero factors in equations, i.e. if a 6= 0 then ab = ac = b = c.
Define a relation

V2000: More Notes for Week 3
Correct versus Useful definitions: Some definitions are correct, but are not
put in a form that helps one build proofs. For example, if f : X Y is a function
and A X then one can define the image f [A] of A to be the set cfw_f

V2000 (Spring 2016): Notes on the project: first report due Thurs March 24
The basic assignment is to write a two page paper that explains some proof. This paper
could be (but does not have to be) written in some version of Latex. I will put a tex file
on

V 2000, HW 9; due Mar 31 in class
Ex 1: from book: Dumas-McCarthy Ex 5.6
Ex 2. Prove from the definition of limit that lim (x2 1) = 3.
x2
Ex 3. Prove from the definition of limit that
x+2
1
lim
= .
x1 x + 3
2
Ex 4. Prove from the definition that limx2 3x

MATH V2000: Review for final, May 2016
There will be eight questions each worth 20 points, of which you should do six. (There is
no advantage to answering more questions; I will grade only 6 of them.) Write complete
proofs justifying all statements. The s

Intro to Higher Math
Practice Midterm II
Instructor: David Hansen
Please explain your reasoning and prove your claims. Each problem is worth 20 points.
1. Prove that 6n 1 is divisible by 5, for any n 1.
Solution: Induction on n. True for n = 1. Assume 5 d

V2000: Notes for Week 3: Operations and equivalence relations
I think that one difficulty with the book (besides its brevity) is that it is not always
completely explicit. You do need to get used to that because mathematics (specially
the elementary stuff