22
4. September 5th: Naive set theory.
4.1. Sets: Basic denitions. Now that we are a little more experienced with
the basic language of logic, we can go back to our denition of set and review several
related notions.
Recall that, for us, a set S is a coll
3. SEPTEMBER 3RD: MORE ABOUT LOGIC. TECHNIQUES OF PROOFS.
15
3. September 3rd: More about logic. Techniques of proofs.
3.1. The behavior or quantiers under negation. Suppose a statement p
depends on a variable and a quantier:
8 x , p( x )
as in 1.5. What
52
9. September 24th: Quotients modulo an equivalence relation.
9.1. Quotient of a set by an equivalence relation. Now that we know about
the link between equivalence relations and partitions, we can give a better look at
the following notion, already men
5. SEPTEMBER 10TH: NAIVE SET THEORY: CARTESIAN PRODUCTS, RELATIONS.
29
5. September 10th: Naive set theory: cartesian products, relations.
5.1. The cartesian product. Towards the end of 1.1, we mentioned briey
that round parentheses are used to denote ord
6. SEPTEMBER 12TH: INDUCTION.
35
6. September 12th: Induction.
6.1. The well-ordering principle and induction. We will come back to set
theory and relations very soon, but rst we take a moment to cover another powerful
technique of proof, which I left out
42
7. September 17th: Proof of the binomial theorem. Equivalence relations
and equivalence classes
7.1. The binomial theoremproof. Recall that at the end of the previous
section we stated the following theorem:
The binomial theorem. For all n 0,
n
X n
n
10. SEPTEMBER 26TH: FUNCTIONS.
57
10. September 26th: Functions.
10.1. Functions. Next we move to another fundamental type of relations: functions. You have used functions extensively in Calculus, and you have drawn many
graphs of real functions of one re
48
8. September 19th: Partitions and quotients.
8.1. Partitions and equivalence classes. A partition of a set A is a splitting
of A into a collection of disjoint subsets:
More precisely:
Definition 8.1. A partition of a set A is a family of subsets of A,
12. OCTOBER 10TH: INJECTIVE, SURJECTIVE FUNCTIONS.
67
12. October 10th: Injective, surjective functions.
12.1. Injections and surjections. You are probably familiar with the denitions that follow. The surprise down the road will be that they have a lot to
MGF3301 practice questions
September 24, 2013
WARNING: These questions are not necessarily too similar to the questions that will be on
your test. Your test may be longer than this.
Express in correct mathematical notation the set of all points (x, y ) R
62
11. October 8th: Functions: Composition, inverses.
11.1. Functions, recap. Here is a reminder of the situation: functions f : A !
B (that is, from a set A to a set B ) are (determined by) certain relations f A B .
We call f the graph of f . Graphs are
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13. October 15th: Isomorphisms
13.1. Isomorphisms of sets. As we observed above, Theorem 12.7 has the
following consequence: a function f : A ! B is an isomorphism (that is, it has a
two-sided inverse) if and only if it is a bijection (that is, both in
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16. October 24th: Dedekind cuts: existence of least upper bounds, and
decimal expansions
Before proving that D (unlike Q) is complete, we can verify that is a total order
in D, that is:
Proposition 16.1. For any two Dedekind cuts (A1 |B1 ) and (A2 |B2
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14. October 17th: Canonical decomposition
14.1. Canonical decomposition. Now that we have acquired some familiarity with injective and surjective functions, we can appreciate a simple but important result. This is the prototype of a series of results k
15. OCTOBER 22TH: DEDEKIND CUTS: DEFINITION AND EMBEDDING OF Q.
87
15. October 22th: Dedekind cuts: denition and embedding of Q.
15.1. Real numbers: general considerations. Our next major task is a better understanding of the notion of innity. The set R o
2. AUGUST 28TH: LOGIC
7
2. August 28th: Logic
2.1. Logical connectives. Sets are specied by spelling out properties satised
by their elements. My examples have been a little sloppy, actually: when I wrote
S = cfw_n even integer | 0 n 8
you could have obje
1. AUGUST 26TH: BASIC NOTATION, QUANTIFIERS
1
1. August 26th: Basic notation, quantiers
1.1. Sets: basic notation. In this class, Sets will simply be collections of
elements. Please realize that this is not a precise denition: for example, we are not
sayi
Quiz #11 (Form A) KEY
1) A proposed amendment to the U.S. Constitution has passed both the House and
the Senate with the required 2/3 super majority. Each state holds a vote on the
amendment and it receives a majority vote in all but 13 of the 50 states.
Quiz #10 (Form B) KEY
ANSWERS ARE IN BOLD ITALICS.
1) Which number is next in the Fibonacci sequence of numbers: 1, 1, 2, 3, 5, 8, 13, 21 . .
a) 55
b) 1.62
c) 8
d) 34
2) Which of the following is NOT an example of Fibonacci numbers found in nature?
a) a m
Quiz #10 (Form A)
KEY
ANSWERS ARE IN BOLD ITALICS.
1) Which number is next in the Fibonacci sequence of numbers: 1, 1, 2, 3, 5, 8, 13, 21 . .
a) 55
b) 34
c) 8
d) 1.62
2) Which of the following is NOT an example of Fibonacci numbers found in nature?
a) spi
Quiz #9 (Form B)
KEY
March 31, 2004
Answers are in bold & italicized.
1) When an object can be shifted, say to the left or right, and still remain the same, it is
said to have:
a) reflection symmetry
b) rotation symmetry
c) translation symmetry
2) The num
Quiz #9 (Form A)
March 31, 2004
KEY
Answers are in bold & italicized.
1) The frequency of middle C is
a) 440 cps
b) 260 cps
c) 520 cps
d) 130 cps
2) The frequency of G one octave below middle G is:
a) double the frequency of middle G
b) four time the freq
Quiz #8 (form A) KEY
Each problem is worth 1 point. You will not receive any partial credit for wrong answers on this quiz.
Everyone who takes the quiz begins with 4 points credit. Some answers have been rounded to the nearest
dollar. ANSWERS IN BOLD ITAL
MGF 1107
Quiz #7 (form B)
KEY
Answers are in bold italics.
1) Consider a typical 30-year fixed-rate mortgage. During which of the following years is the highest portion of each
payment applied toward principal?
a) First year
b) Tenth year
c) Twentieth yea
MGF 1107
Quiz #7 (form A)
KEY
Answers are in bold italics.
1) Consider a typical 30-year fixed-rate mortgage. During which of the following years is the highest portion of each
payment applied toward interest?
a) First year
b) Tenth year
c) Twentieth year
Quiz 6 Key
OCT. 15, 2004
1.) Dan deposits $300 each month into a retirement fund that pays 2.75% compounded monthly. How
much will he have in the account when he retires after working for 35 years?
PMT = 300
APR = 0.0275 n = 12 Y = 35
12 35
A = 300
(1 +
Quiz #6 (form A) KEY
Oct. 15, 2004
1.) Dan deposits $200 each month into a retirement fund that pays 3.75% compounded monthly. How
much will he have in the account when he retires after working for 35 years?
PMT = 200
APR = 0.0375 n = 12 Y = 35
12 35
A =
Quiz #5 (Form B) KEY
Oct. 8, 2004
1.) You invest $6,250 in a saving account for a period of 3 years. How much money will you end up with
if your account pays 5.25% compounded . . .
a)
Quarterly? (2 points)
P=6250
Use:
APR = 0.0525
APR
A = P 1 +
n
n=4
y=3
Quiz #4 (Form B)
KEY Answers are in bold and italics.
1) Given the following set of test scores:
find the following:
a) Range
92
85
63
50
78
96
88
66
94 92
96 50 = 46
b) standard deviation
15.7
c) Estimate the standard deviation using the range rule of th
Quiz #4 (Form A) KEY
Answers are in bold and italics.
1) Given the following set of test scores:
92 85 63 57
find the following:
a) Range
78
96
88
66
94 92
96 57 = 39
b) standard deviation
14.3
c) Estimate the standard deviation using the range rule of th