EXAM 1 SOLUTIONS
Problem 1. If |A| = 4 and |B | = 5, then |A B | = 9. Proof. This is false. The product has size 4 5 = 20. It is straightforward to construct a counter-example if you want. Problem 2.
THINGS TO KNOW FOR EXAM 3
4. Theorems to know
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)
(1) (2) (3) (4)
(5) (6) (7) (8)
Be able to prove the theorems indicated by a below. 1. D
THINGS TO KNOW FOR EXAM 2
3. Theorems to know Be able to prove all of the theorems indicated below. Theorem 6.2 (pg. 130) Theorem 8.2 (pg. 181) Theorem 8.3 (pg. 182) Theorem 8.4 (pg. 183) Theorem 8.6
REVIEW SHEET FOR EXAM 1
(1) Sets (a) Empty set (b) Subsets (c) Notation for sets (page 15) (d) Indexed collections of sets. (e) Partitions of sets. (f) Set operations. (g) Power sets. (h) Sets as elem
RED
Name: Instructor: Pace Nielsen
Math 290 Sample Exam 3
Note that the rst 10 questions are true-false. Mark A for true, B for false. Questions 11 through 20 are multiple choicemark the correct answe
RED
Name: Instructor: Pace Nielsen
Math 290 Sample Exam 2
Note that the rst 10 questions are true-false. Mark A for true, B for false. Questions 11 through 20 are multiple choicemark the correct answe
RED
Name: Instructor: Darrin Doud
Math 290 Sample Exam 1
Note that the rst 10 questions are true-false. Mark A for true, B for false. Questions 11 through 20 are multiple choicemark the correct answer
LECTURE 31: EUCLIDEAN ALGORITHM (11.4-5)
The only way of nding the limits of the possible is by going beyond them into the impossible. Arthur C. Clarke (1917 - ) 1. Using the Euclidean Algorithm to fi
LECTURE 30: GREATEST COMMON DIVISORS (11.3-4)
The greatest use of life is to spend it for something that will outlast it. William James (1842 - 1910) 1. The Division Algorithm Examples From last time:
LECTURE 29: DIVIDING (11.1-2)
Nothing is particularly hard if you divide it into small jobs. Henry Ford (1863 - 1947) 1. Division facts we will use over and over Here are some denitions weve seen befo
LECTURE 28: INJECTIONS AND BIJECTIONS (10.5)
Things are only impossible until theyre not. Jean-Luc Picard 1. Genealogy and comparing cardinalities Recall that |A| |B | means there is an injection from
LECTURE 27: CARDINALITY OF POWER SETS (10.4)
Most powerful is he who has himself in his own power. Seneca (5 BC - 65 AD) 1. Comparing Cardinalities Intuitively, we know that |N| should be smaller than
LECTURE 26: UNCOUNTABLE SETS (10.3)
When you encounter diculties and contradictions, do not try to break them, but bend them with gentleness and time. Saint Francis de Sales (1567 - 1622)
1. Uncountab
LECTURE 25: DENUMERABLE SETS (10.1-10.2)
We have too many high sounding words, and too few actions that correspond with them. Abigail Adams 1. Review from last time Two sets have equal cardinality, or
LECTURE 24: CARDINALITY (10.1-10.2)
We have too many high sounding words, and too few actions that correspond with them. Abigail Adams 1. Cardinality Revisited We have used |A| to denote the number of
LECTURE 23: INVERSES AND PERMUTATIONS (9.6-9.7)
Let all things be done decently and in order. -Paul the apostle 1. More on inverses From last time: Theorem 1. Let f : A B be a function. The inverse re
LECTURE 22: COMPOSITION AND INVERSES (9.5-9.6)
There is no doubt that the rst requirement for a composer is to be dead. Arthur Honegger (1892 1955) (Ask for students to write down topics they would li
LECTURE 21: MATCHING SETS (9.3-9.4)
2 is not equal to 3, not even for large values of 2. Grabels Law 1. One-to-One Recall that a relation from A to B is a function if each element of A occurs exactly
LECTURE 20: FUNCTIONS (9.1-9.2)
Success is more a function of consistent common sense than it is of genius. An Wang 1. What are functions? Let A and B be two non-empty sets. A function f from A to B ,
LECTURE 19: THE INTEGERS MODULO n (8.6)
Appreciation is a wonderful thing: It makes what is excellent in others belong to us as well. Voltaire (1694 - 1778) 1. Tables Recall that the relation on Z giv
LECTURE 18: CONGRUENCE MODULO n (8.5)
In battling evil, excess is good; for he who is moderate in announcing the truth is presenting half-truth. He conceals the other half out of fear of the peoples w
LECTURE 17: PROPERTIES OF EQUIVALENCE RELATIONS (8.4)
Mi taku oyasin. (We are all related.) Lakota belief 1. Equivalence classes Let R be an equivalence relation on A. Let a A. The set [a] = cfw_x A :
LECTURE 16: EQUIVALENCE RELATIONS (8.1-8.3)
Relationships of trust depend on our willingness to look not only to our own interests, but also the interests of others. Peter Farquharson He that ndeth hi
LECTURE 15: INDUCTION STRENGTHENED (6.4)
I think I should have no other mortal wants, if I could always have plenty of music. It seems to infuse strength into my limbs and ideas into my brain. Life se
LECTURE 14: INDUCTION GENERALIZED (6.2)
There are two kinds of people, those who nish what they start and so on. Robert Byrne 1. Generalization Induction works if we start our rst step anywhere in Z.
LECTURE 13: INDUCTION (6.1)
Courage is an accumulation of small steps. George Konrad 1. Another Axiom Denition 1. If S N is a subset of the natural numbers, and S = , then S has a minimal element. Thi
LECTURE 12: PROVE OR DISPROVE (7.3)
The reverse side also has a reverse side. Japanese Proverb 1. Testing Statements Sometimes statements are given for which we do not immediately know whether they ar
LECTURE 11: EXISTENCE PROOFS AND EXTRA STUFF (5.3-5.5)
Forming characters! Whose? Our own or others? Both. And in that momentous fact lies the peril and responsibility of our existence. Elihu Burritt
LECTURE 10: COUNTER-EXAMPLES AND CONTRADICTIONS (5.1-5.2)
Great spirits have always encountered violent opposition from mediocre minds. Albert Einstein (1879 - 1955) 1. Counter-examples We have often
LECTURE 9: PROOFS FOR REAL NUMBERS AND SETS (4.3-4.5)
The human race has one really eective weapon, and that is laughter. Mark Twain (1835 - 1910) 1. Inequalities We can assume some basic facts about