1. AUGUST 26TH: BASIC NOTATION, QUANTIFIERS
1
1. August 26th: Basic notation, quantiers
1.1. Assume that P (x) is some statement whose truth depends on x. Which of
the following is grammatically correct ?
P (x), 8
8x, P (x)
8x 2 P (x)
Answer: To x idea
11. OCTOBER 8TH: FUNCTIONS: COMPOSITION, INVERSES.
31
11. October 8th: Functions: Composition, inverses.
11.1. Let f , g denote functions R ! R.
Assume (8x) f (x) = x 1. Does there exist a function g such that g f (x) =
x2 2x? Find one, or prove that no
15. OCTOBER 22ND: DEDEKIND CUTS: DEFINITION AND EMBEDDING OF Q.
43
15. October 22nd: Dedekind cuts: denition and embedding of Q.
15.1. Prove that the function Q ! D dened in 15.3 is injective: that is, prove
that if q1 6= q2 , then cfw_x 2 Q | x < q1 = c
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16. October 24th: Dedekind cuts, II.
16.1. With notation as in the proof of Theorem 16.2, prove that x < y for all
x 2 L and all y 2 M .
Answer. Since x 2 L (and by denition of union over families, Denition 4.7)
there exists an index j 2 I such that x
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13. October 15th: Isomorphisms.
13.1. Let F be a family of sets. Prove that the isomorphism relation denes
=
an equivalence relation on F . That is, show that (for all sets A, B, C in F )
A A;
=
If A B, then B A;
=
=
If A B and B C, then A C.
=
=
=
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18. October 31st: Cardinality II
18.1. Prove that the notation |A| < |B| introduced in 18.1 is well-dened.
Answer: Well-dened means that the notion is independent of the representatives. So we have to verify that if A0 A, B 0 B, and A, B satisfy the co
50
19. November 5th: Cardinality III
19.1. Prove that any nite product A1 Ar of Q
countable sets is countable.
Question: is a countable product of countable sets, n2N An , necessarily countable?
Answer: We can view A1 Ar
2
Ar
1
Ar as
( (A1 A2 ) A3 ) A4
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20. November 7th: Cardinality IV
20.1. Let A, B be sets. Prove that |A| |B| if and only if there exists a surjective
function B ! A. (Argue as in the beginning of the proof of Proposition 18.4.)
Answer: By denition, |A| |B| if and only if there is an i
21. NOVEMBER 19TH: TOPOLOGY I
55
21. November 19th: Topology I
21.1. Verify formally that limx!1 x2 = 1.
Answer: We need to verify that
Given , let
8 > 0 , 9 > 0 : 0 < |x
=
min( 3 , 1).
If
|x
then in particular |x
But then we have |x
=) |x2
1| <
1| < min
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23. November 26th: Topology III
23.1. Let (T, U ) be a topological space, and let S T be a subset. Verify that
the collection of sets U \ S, as U 2 U , forms a topology.
Answer: We have to verify that the family V = cfw_U \ SU 2U satises the four
prope
24. DECEMBER 3RD: TOPOLOGY IV
61
24. December 3rd: Topology IV
24.1. Prove that R with the Zariski topology is connected.
Answer: Let S R be any set other than ; and R. Then S and R r S cannot
both be nite, since otherwise R itself would be nite. It follo
Introduction to Advanced Mathematics Midterm I
October 3, 2013
Where appropriate, write your answers in the space provided after each question. Please make an
effort to write neatly.
Your Name: KEY
1
Intentionally left blank
2
(1) Basic notation, logic, a
8. SEPTEMBER 19TH: EQUIVALENCE RELATIONS AND PARTITIONS.
23
8. September 19th: Equivalence relations and partitions.
8.1. Describe the subset of AA corresponding to the relation =. Draw its picture
in R2 , for the case A = R.
Answer: In general, the subse
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10. September 26th: Functions.
10.1. For every set A, there is exactly one function f : A ! A whose graph
determines an equivalence relation on A. What is this function?
Answer: This function is the identity. Indeed, let be the graph of f . Since is
an
5. SEPTEMBER 10TH: NAIVE SET THEORY: CARTESIAN PRODUCTS, RELATIONS.
15
5. September 10th: Naive set theory: cartesian products, relations.
5.1. If S is a nite set, with |S| = n elements, how many elements does the power
set P(S) contain?
Answer: |P(S)| =
6. SEPTEMBER 12TH: INDUCTION.
17
6. September 12th: Induction.
6.1. An order relation on a set S is a total ordering if for every two elements
a and b in S, either a b or b a.
Give an example of an order relation that is not total.
Give an example of a
2. AUGUST 28TH: LOGIC
3
2. August 28th: Logic
2.1. Prove De Morgans laws.
Answer: Recall that these say
(p _ q) () (p) ^ (q)
We can take care of these
p q
T T
T F
F T
F F
(p ^ q) () (p) _ (q) .
with
p
F
F
T
T
two truth tables:
q p _ q (p _ q) (p) ^ (q)
F
4. SEPTEMBER 5TH: NAIVE SET THEORY.
11
4. September 5th: Naive set theory.
4.1. Prove parts (7) and (8) of Theorem 4.10.
Answer: These statements are
S r (A \ B) = (S r A) [ (S r B)
S r (A [ B) = (S r A) \ (S r B) .
They are set-theoretic incarnations of
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7. September 17th: Binomial theorem. Equivalence relations and classes.
7.1. Explain why adding up every other number on a line of Pascals triangle gives
a power of 2. For example, from the line for n = 6:
1 + 0 + 15 + 0 + 15 + 0 + 1 = 32 = 25
.
(Use t
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12. October 10th: Injective, surjective functions.
12.1. Suppose A and B are nite sets with the same number of elements, and
that f : A ! B is an injective function. Prove that f is also surjective.
Now suppose that A and B are both innite sets, and t
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17. October 29th: Cardinality I
17.1. Given two nite sets A, B, construct another nite set M such that |M | =
|A| |B| by only invoking set-theory operations and without mentioning numbers
in any way. Then do the same for +: construct a set S such that
14. OCTOBER 17TH: CANONICAL DECOMPOSITION.
41
14. October 17th: Canonical decomposition.
e
14.1. Let f : A ! B, : A ! A/, : im f ! B, f : A/ ! im f be as in
this section. Since is injective, it has a left-inverse ; since is surjective, it has a
e
right-in
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22. November 21st: Topology II
22.1. Let Rstan , resp. RZar be the topological spaces determined by the standard,
resp. Zariski topology on R. Prove that the identity R ! R is a continuous function
Rstan ! RZar , and that it is not a homeomorphism.
Ans
8
3. September 3rd: More about logic. Techniques of proofs.
3.1. The fact that (8x : p(x) is logically equivalent to (9x : p(x) may evoke
one of De Morgans laws: (p ^ q) is logically equivalent to (p) _ (q). Find out in
what sense the latter is a particul
9. SEPTEMBER 24TH: QUOTIENTS MODULO AN EQUIVALENCE RELATION.
25
9. September 24th: Quotients modulo an equivalence relation.
9.1. Consider the relation on R2 r cfw_(0, 0) dened by declaring that (x1 , y1 )
(x2 , y2 ) if and only if 9r 2 R>0 , (x2 , y2 )
Introduction to Advanced Mathematics Midterm II
November 14th, 2013
Write your answers in the space provided after each question. Please make an effort to write neatly.
Your Name: KEY
1
For your convenience, I am indicating places in the class notes where