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. Let x R. The sentence If x > 3, then x2 < 2. is a state
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. Definitions (1) Theorem 10.1 (2) Result 10.2 equal cardi
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 (pg. 185) Theorem 8.9 (pg. 191) Theorem 9.4 (pg. 203) T
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 elements of other sets. (i) Dierence between elements and s
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 answer on your bubble sheet. Answers to the last ve question
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 answer on your bubble sheet. Answers to the last ve question
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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 on your bubble sheet. Answers to the last ve questions
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 find linear combinations Lets nd the gcd of 27 and 17. We
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: Theorem 1. Given integers n, d Z (called the numerator
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 before. Denition 1. Let a, b Z. 1. We say that a divides b,
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 A to B . From the notation, we might expect that if |A
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 |R|. This is how can we make this precise: Denition 1.
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. Uncountable sets One way to think about uncountable sets is the
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 equal size, if there is a bijection between them. We s
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 elements in A. But for innite sets this is not precise
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 relation f 1 is a function from B to A if and only if f i
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 like covered more deeply.) 1. Identity Function Let A be
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 once as a rst coordinate in the relation. We are also i
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 , written f : A B , is a relation (i.e. a subset of A 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 given by a b (mod n) is an equivalence relation. The equiv
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 wrath. Kahlil Gibran (1883 - 1931) Theorem 1. Let n Z wi
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 : xRa is called the equivalence class of a. This is the
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 his life shall lose it: and he that loseth his life for m
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 seems to go on without eort, when I am lled with music. G
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. Result. For every nonnegative integer n we have 2n > n.
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. This fact is called the well-ordering property for the nat
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 are true or false. We can test them, and try to do a proo
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 1. Quick Review Suppose we are trying to prove a statem
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 been dealing with proving universal statements like R :
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 inequalities. Let a, b R. We know a2 0. Also, if a 0 th