Introduction to Advanced Mathematics Midterm II
November 3rd, 2011
Write your answers in the space provided after each question. Please make an effort to write neatly.
The problems marked by refer directly to material in the class notes or your homework.

Introduction to Advanced Mathematics
Course notes
Paolo Alu
Florida State University
1. AUGUST 30TH: BASIC NOTATION, QUANTIFIERS
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1. August 30th: Basic notation, quantiers
1.1. Sets: basic notation. In this class, Sets will simply be collections of
eleme

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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

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26. December 6th Topology II.
26.1. I have allowed the endpoints of the open intervals considered in Denition 26.5 to be 1. Prove that it is not necessary to do this, in the sense that one
denes the same topology by just taking unions of bounded open i

14. OCTOBER 16TH: FUNCTIONS: EXAMPLES. COMPOSITION, INVERSES.
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14. October 16th: Functions: examples. Composition, inverses.
14.1. Let f : A ! B be a function. Prove that the set of nonempty inverse images
of f forms a partition of A.
Answer: I will giv

13. OCTOBER 11TH: FUNCTIONS.
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13. October 11th: Functions.
13.1. Let A, B be sets, and f : A ! B a function. Let S and T be subsets of A.
Is f (S \ T ) necessarily equal to f (S ) \ f (T )? Give a proof or nd a counterexample.
Is f (S [ T ) necessaril

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12. October 9th: Quotients modulo an equivalence relation.
12.1. How many dierent equivalence relations are there on a set with 4 elements?
Answer: 15. Indeed, by Theorem 11.6 and Theorem 12.1, counting equivalence
relations is equivalent to counting p

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11. October 4th: Partitions. More about equivalence classes.
11.1. Describe explicitly the equivalence classes for the relation dened in Exercise 10.2. How many distinct equivalence classes are there? (This gives a description
of the set Z/ in this cas

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10. October 2nd: Equivalence relations and equivalence classes.
10.1. Describe the subset of A A corresponding to the relation =. Draw its
picture in R2 , for the case A = R.
Answer: In general, the subset corresponding to a relation consists of the pa

9. SEPTEMBER 25TH: THE BINOMIAL THEOREM.
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9. September 25th: The binomial theorem.
9.1. Let C (n, k ) be the number of subsets with k elements of a set with n elements.
Prove that C (n, k ) = n . (Hint: First prove that C (n + 1, k ) = C (n, k 1) + C (n

8. SEPTEMBER 20TH: INDUCTION.
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8. September 20th: Induction.
8.1. Prove by induction, or disprove, that 10n
1 is a multiple of 3 for all n
0.
Answer: The stated fact is true. Indeed:
It is true for n = 0, since 100 1 = 0 = 3 0 is a multiple of 3;
We a

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7. September 18th: Relations.
7.1. Let A be a set consisting of exactly 4 elements. How many dierent relations
on A are there?
Answer: The Cartesian product A A has 4 4 = 16 elements. A relation on A
is a subset of A A (Denition 7.8), so counting relat

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6. September 13th: More naive set theory.
6.1. Prove that (A \ B ) \ C = A \ (B \ C ) for all sets A, B , C .
Answer: Let p, q , r respectively be the statements x 2 A, x 2 B , x 2 C . By
denition of intersection, x 2 (A \ B ) \ C means x 2 (A \ B ) ^

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5. September 11th: Techniques of proof. Naive set theory.
5.1. Explain why the truth table for =) tells us that if p is a statement that
implies a false statement, then p is itself false.
Answer: The truth table for p =) q is
p
T
T
F
F
p =) q
T
F
T
T
q

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4. September 6th: More examples.
4.1. Are
and
8x > 0 9N : n > N =) |sn | < x
8x > 0 9N : n > N =) |sn | < 3x
equivalent statements? If so, prove that they are. If not, nd an example of an
assignment sn which satises one of the statements but not the ot

1. AUGUST 26TH: BASIC NOTATION, QUANTIFIERS
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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

2. AUGUST 28TH: LOGIC
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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

3. SEPTEMBER 3RD: MORE ABOUT LOGIC. TECHNIQUES OF PROOFS.
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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

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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.
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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.
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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

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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.
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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

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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.
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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

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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