Math 290 Lecture #9
4.3-4.4: More on Proofs, Part II
4.3: Proofs Involving Real Numbers. We start by recalling some basic properties
of the real numbers, things that we will take as true.
For a R and n N, we have an 0 when n is even, and an < 0 when a < 0
Math 290 Lecture #24
10.1,10.2: Cardinalities of Sets
The cardinality |S| of a set S is the number of elements in S.
We call S a nite set if S = or |S| N, while we call S an innite set if S is not a
nite set.
We might be tempted to write |S| = when S is a
Math 290 Lecture #26
10.3: Uncountable Sets, Part II
In review, we have several possibilities for the cardinality of a set A:
(a) A = , so that |A| = 0;
(b) A is a nite set so that |A| N;
(c) A is an countably innite or denumerable set so that |A| = |N|;
Math 290 Lecture #23
9.6,9.7: Functions, Part IV
9.6: Inverse Functions. Every relation R A B has an inverse relation R1 from
B to A dened by
R1 = cfw_(b, a) : (a, b) R.
Example. For A = cfw_1, 2, 3 and B = cfw_4, 5, the inverse of
R = cfw_(1, 4), (2, 4),
Math 290 Lecture #27
10.4: Comparing Cardinalities of Sets
We have seen that the innite sets N and R are not numerically equivalent, that they
represent two dierent kinds of innity.
We think that |N| is smaller than |R| and would like to write |N| < |R|.
Math 290 Lecture #28
10.5: The Schrder-Bernstein Theorem
o
For sets A and B we have seen the notions of |A| = |B and |A| |B| and |A| < |B|.
For real numbers a and b we know that if a b and b a, then a = b.
Could we say that if |A| |B| and |B| |A|, then |A
Math 290 Lecture #29
11.1,11.2: Proofs in Number Theory
11.1: Divisibility Properties of Integers. Recall that an integer p 2 is prime
if it only positive integer divisors are 1 and p.
The rst few prime numbers are 2, 3, 5, 7, 11, and 13.
How many even pr
Math 290 Lecture #31
11.5: Relatively Prime Integers
We have learned that d = gcd(a, b) is the smallest positive integer of the form d = as + bt
and also how to compute d and s and t.
Now we turn our attention to another question: if a, b, c are integers
Math 290 Lecture #30
11.3,11.4: Greatest Common Divisors and Euclids Algorithm
11.3: Greatest Common Divisors. What is the largest integer that divides both
12 and 20?
Well each of 2 and 4 divides both 12 and 20, but none of 3, 5, 6, 7, etc, divide both 1
Math 290 Lecture #32
11.6,11.7: More Number Theory
11.6: The Fundamental Theorem of Arithmetic. The prime numbers occupy
a special place in the divisibility of integers.
We can express the number 420 as the product 2 2 3 5 7.
That we can write every integ
Math 290 Lecture #25
10.3: Uncountable Sets
We recall some facts about decimal expansions of real numbers.
Every irrational number has a nonrepeating decimal expansion that is unique:
2 = 1.414 . . . .
Every rational number has a repeating decimal expansi
Math 290 Lecture #22
9.5: Functions, Part III
For functions f : R R and g : R R we can form the sum of f with g by
(f + g)(x) = f (x) + g(x)
and the product of f with g by
(f g)(x) = f (x)g(x).
What makes this sum and product possible is that in the commo
Math 290 Lecture #19
8.5,8.6: Modulo n
8.5: Congruence Modulo n. Recall for a, b, n Z with n 2, that a b mod n
when n | (a b).
This congruence modulo n is a relation R on Z where a R b when a b mod n.
Theorem 8.6. Let n Z with n 2. The relation R of congr
Math 290 Lecture #13
6.1: Induction Part I
We now consider another way to prove a quantied statement x S, P (x) when S is a
well-ordered set.
We will get to what we mean by well-ordered in a minute.
An element m of a nonempty A R is called a least element
Math 290 Lecture #14
6.2: Induction Part II
There are times when we have quantied statements like n cfw_m, m+1, m+2, . . . , P (n)
for some integer m that we would like to prove using induction.
It should not come as a surprise that a principle of mathema
Math 290 Lecture #16
7.3: Testing Statements
For a given statement whose truth value is not know (to us), we can determine whether
it is true or false, and we should be able to back this up by proving or disproving it.
Example 7.8. For sets A, B, and C, d
Math 290 Lecture #15
6.3,6.4: Induction Part III
6.3: Proof by Minimum Counterexample. Sometimes induction does not work
to well in showing that n N, P (n) is true.
If we attempt to prove n N, P (n) by contradiction, we would begin by assuming that
n N, P
Math 290 Lecture #18
8.3,8.4: Equivalence Relations and Classes
8.3 Equivalence Relations. A familiar relation is that of equality on Z.
The relation R of equality is dened by a R b if a = b.
This relation is reexive because a R a, i.e., a = a, for every
Math 290 Lecture #12
5.4,5.5: More on Proofs, Part V
5.4: Existence Proofs. A theorem about existence of some x S for which R(x) is
true has the logical form
x S, R(x).
We might be able to nd or construct an actual value of x S for which R(x) is true.
The
Math 290 Lecture #17
8.1,8.2: Relations and Their Properties
8.1: Relations. There are many examples, as we shall see, of relations in mathematics.
A relation R from a set A to a set B is a subset R of A B.
We say that a A is related to b B when (a, b) R,
Math 290 Lecture #20
9.1,9.2: Functions, Part I
You have encountered functions in precalculus and calculus courses.
We will look at functions in a more in depth fashion in this Chapter.
9.1: The Denition of a Function. For nonempty sets A and B, a functio
Math 290 Lecture #21
9.3,9.4: Functions, Part II
9.3: One-to-One and Onto Functions. There are two important properties that
functions may possess.
A function f : A B is called one-to-one or injective if every two distinct elements of
A have distinct imag
Math 290 Lecture #35
12.4: Fundamental Properties of Limits of Functions
No one wants to keep constructing proofs to justify the limits of functions.
It is better to have a higher-level way of computing limits of functions, and this is
precisely what we w
Math 290 Lecture #33
12.1: Limits of Sequences
We recall the notion of a sequence from Calculus, and review the idea of limits in terms
of a sequence.
A sequence of real numbers is a function f : N R.
In our notation for a function a sequence has the form
Math 290 Lecture #36
12.5: Continuituy
For X R, let f : X R be dened in some deleted neighbourhood of a R (where it
could be dened at a too).
Recall that limxa f (x) = L means that for every
|f (x) L| < whenever 0 < |x a| < .
> 0 there exists > 0 such tha
Math 290 Midterm 2 Review
Sections 2 and 4
Midterm exam 2 is in the Testing Center on Monday, Mar 1 and Tuesday, Mar 2. There is no
late day.
The exam will emphasize the topics in Chapters 4, 5, 6, and 8. (Sections 8.5 and 8.6 will not be
on this exam.) T
NAME
Math 290 Midterm Exam 2
Sections 2 and 4 Winter 2010
Monday, Mar 1 through Tuesday, Mar 2 Testing Center
Professor: David Cardon, 302 TMCB, Campus Ext. 2-4863
Instructions:
There are 20 questions for a total of 100 points.
Questions 112 are true-fa
Math 290 Midterm 3 Review
Sections 2 and 4
Midterm exam 3 is in the Testing Center on Wed, Mar 24 and Thursday, Mar 25. There is no
late day.
Sections covered on the exam: 8.48.6, 9.19.7, 10.110.4.
The exam questions will be similar to textbook examples a
NAME
Math 290 Midterm Exam 3
Sections 2 and 4 Winter 2010
Wed, Mar 24 through Thu, Mar 25 Testing Center
Professor: David Cardon, 302 TMCB, Campus Ext. 2-4863
Instructions:
There are 25 questions for a total of 100 points.
Questions 120 are true-false a
Math 290 Midterm Exam 3
True-False and Multiple Choice Section: Questions 120
1. Let P = cfw_A : I be a partition of a nonempty set A. Then there exists an equivalence
relation R on A such that P is the set of equivalence classes determined by R.
(a) True