Math 431

Thought of the week:

The Math department is great!

Axiom 1: If R is the union of the non-empty left ray A and the non-empty right ray B, and A and B do not intersect, then either A has a largest element or B has a smallest element.

Axiom 2: (L.U.B.[Least upper bound]): Suppose A is a set with the property there is a number b such that if x € A, then x <= b then there is a number q such that if x € A then x <= q and if y<q then there is a z € A such that z > y.

open interval: (a,b)={x:x is a number and a<x and x<b}
half-open interval: [a,b)={x:x is a number and a<x and x<b}
half-open interval: (a,b]={x:x is a number and a<x and x<b}
closed interval: [a,b]={x:x is a number and a<x and x<b}

Nickname	Definitions
Left/Right ray	Def. The number set, A, is a right ray means that if x is in A and y > x, then y is in A. We similarly define a left ray.
Limit point	Def. The statement that the number, p, is a limit point of the number set, A, means that if (a,b) is an open interval containing p, (that is: a<p<b) then there is a number q such that a<q<b, q is in A, and q does not equal p. A more concise statement of the previous definition may read: P is a limit point of A if and only if each open interval containing p contains a number in A different from p.
æ Limit point	Def. The statement that the number, p, is a limit point of the number set, A, means that if æ > 0 then there is at least one point q that is an element of A, 0 < \|q - p\| < æ
Closed set	Def. The statement that the number set, A, is closed means that if p is a limit point of A then p is in A.
Open set	Def. The statement that the number set, A, is open means that if p is in A then there is an open interval containing p which is contained in A.
Function	Def. The statement that f is a function means that f is a collection, each member of which is an ordered pair, no two of which have the same first coordinate. The set of first coordinates for f is called the domain of f, while the set of second coordinates is called the image of f.
Sequence	Def. The statement that S is a sequence means that S is a function with domain some initial segment of the positive integers. (That is: the domain of S is either the set of positive integers or the domain of f is the set {1,2,3,...,n} for some positive integer n.)
Limit of a sequence	Def. The statement that p is the limit of the sequence S means that if (a,b) is an interval containing p, then there is a positive integer N such that S(i) is in (a,b) for each positive integer i>N.
æ Limit of a sequence	Def. Sequence S has limit L means for each æ > 0 there exists a n € s.t. if i > n then \|S(i) - L\| < æ
Sub-sequence	Def. The statement that T is a sub-sequence of the sequence S means there is an increasing sequence, I, of positive integers such that T=S(I).
Countable	Def. The statement that the set A is countable means there is a surjective . f: Z -->A
Surjective	Def. f:D --> H is surjective/onto means if x € H then there is a d € D such that f(d) = x
Infinite	Def. The statement that A is infinite means if n € Z+ then there exists a C which is a subset of A with n elements
Monotonically increasing/ Monotonically non-decreasing	Def. The function f: A --> B is monotonically increasing means if x, y € A, x<y, then f(x) < f(y) Similarly monotonically non-decreasing means if x<y, x,y € A then f(x) <= f(y)
Dense	Def. The statement that the number set A is dense in R means that if x € R then x is a limit point of A.
Closure	If A is closed, then Ã = {x: x € A or x is a limit point of A} is called closure of A.
Nowhere Dense	B is nowhere dense means if p € (a,b), there is (c,d) such that (c,d) not intersect B. Or simply closure contains no open intervals
Continuous at p	The statement that f: A --> B is continuous at p € domain(f) means if f(p) € (a, b) there is a (c,d) containing p such that if x € (c,d) intersect the domain(f) then f(x) € (a,b)
æ/delta continuous at p	The statement f:A --> B is continuous at p means if æ > 0, then there exists a delta > 0 s.t. if x € dom(f) and 0 < \| x - p\| < delta, then 0 < \| f(x) - f(p) \| < æ.
onto	F:A --> B is onto (surjective) means if y € B, there exists a eee A such that f(a) = b
1-1	f:A --> B is 1-1 (injective) means if f(a sub 1) = f(a sub 2) then a sub 1 = a sub 2
subset	If D is a subset of R, then the statement that U is open (closed) relative to D means there is an open(closed) set V such that U is the intersection of V with D.
Seperable

Problems:

#	Finished by	Scary Problems
T	HH,VM	Can you find a method to count the number of tiles in an infinitely large tiled room?
P	AH	When a plane takes off is there a First moment when the plane is in the air, a last moment when the plane is on the ground, neither or both?
1	VM,CK	There is a number set, A, such that 0 is a limit point of A.
2	DG	Find a positive (without using the words none or no) statement that means that the set A is infinite.
3	AR	If A is a number set and A has a limit point, p, then A is infinite.
4	VM, RJ, TO	If p is a limit point of A, then p is in A. Shown to be False
5	RJ, MG, AR	If A is infinite, then A has a limit point. Shown to be False
5a	DG	If A is uncountable, then A has a limit point.
5b	DG	Is it possible for an uncountable set to have only one limit point or finitely many?
6	CT, VM	If A is infinite and bounded, then A has a limit point.
7	CT	The closed interval [0,1] is infinite.
8	DG	If c > 0 there is a positive integer N such that (1/N) < c.
9	VM,	There is a number set, A, which has the property that A contains no open interval and each point in A is a limit point of A.
10	AR	There is a closed number set, B, which satisfies the properties of A in problem 9.
11	WW	There is a number, p, such that p*p=2.
12	CT,RJ	There is a number set that is neither open nor closed.
12b	VM,	There is a number set that is both open and closed.
13	DG	There is a number set, other than R, which is both open and closed. Shown to be False
14	HH,AR	The rational numbers in [0,1] form a countable set.
15	VM, TO	There is a set with exactly one limit point.
16	DG	The statement that p is not a limit point of A means?
17	CK	Suppose A is a left ray then B = R - A is a right ray.
18	AR, TO, DG	Find an equivalent definition for limit point that deals with distance (æ > 0), and prove the two definitions are equivalent.
19	TO	Show Axiom 1 is equivalent to Axiom 2.
20	WW,VM	Suppose A != R and B = R - A, A is open iff B is closed.
21a	RJ	If each of A and B is open/closed then A union B is open/closed and A intersect B is open/closed assuming there exists a x € A intersect B.
21b	VM	Suppose for each i € Z+, A sub i is open and A sub i != R, the union of all i A sub i is open, the intersection of all i B sub i is closed.
22a	CK, VM	Is there a sequence S such that {S(n): n € Z+} has exactly 2 limit points?
22b	CK,VM,RJ,DG	Is there a sequence S with limit p such that the limit of S has 2 limit points.
23	CK	There is a sequence with 2 limits. Shown to be False
24a	MG, VM,AR	There is a sequence with no limit.
24b	RJ	Is there a sequence with no limit such that the image of S is contained in [0,1].
24c	HH,VM	Is there a sequence with no limit such that the image of S is infinite?
25	AR,DG	There is a sequence with limit L, such that image(s) has no limit points.
25b	CK, VM,DG	Is there a sequence S such that S has the following properties: 1) S has limit 0 2) S(i) = 0 for infinitely many i 3) S(i) != 0 for infinitely many i
26	CK, VM	Let S be a monotonically non-decreasing sequence such that S(n) < 1 for each n € Z+, then S has a limit.
27	WW, VM, AR	[0,1] is uncountable.
28		Suppose for each i € Z+, A sub i is equal to (a sub i, b sub i) which is an open interval. Suppose for each x in [0,1] there is a positive integer I such that x is in the interval (a sub i, b sub i). Then there exists a finite sub-sequence T sub 1 ... (dot dot dot) T sub N, such that if x is in [0,1] then there exists an integer i < N + 1, such that x is in (A sub T sub i, B sub T sub i). (i.e. Each open cover of [0,1] contains a finite sub-cover)
29	CT	Suppose for each i € Z+, A sub i is a closed sub-set of [0,1] such that A sub one is contained in A sub 2 is contained in A sub 3... (dot dot dot). Then there exists p such that p € (the intersection of A sub i).
29b	CT	Suppose for each i € Z+, A sub i is an open subset of [0,1] such that A sub one is contains A sub 2 contains A sub 3... (dot dot dot). Then there exists p such that p € (the intersection of A sub i). Shown to be false
30	VM	Ã is closed? (Closure of A is closed)
31	WW	Suppose for each i € Z+, A sub i is open and dense in R. Then the intersection of all A sub i from (i = 1 ... infinity) is dense in R.
31b		There is a sequence A sub i of open dense sub sets in R, such that the intersection of all A sub i is not open.
32	WW	Suppose for each i € Z+, B sub i is closed and nowhere dense. Then the union of all B sub i != R.
33	VM	Find an equivalent æ/delta definition for continuity.
34	TO,AR	Suppose f = {(1,1), (2,0)} Is f continuous?
35	CK	Suppose f(x) = 1/x. Is f continuous?
36	CK,VM,AR	Is there a function f:[0,1] --> R such that f is continuous everywhere except 1/2?
37	VM	Is there a function f:[0,1] --> R such that f is continuous only at x = 0?
38	(a)TO, (a)(b)RJ	Suppose each of f & g is continuous at p a) Show f + g is continuous at p b) f * g is continuous at p c) f/g is continuous at p provided g(p) !=0 d) c*f is continuous at p for each c € R
39	CK	Suppose g is continuous at p & f is continuous at g(p). Show f composed with g is continuous at p.
40	CT,VM	Is there a function f:[0,1] --> R such that f is continuous at infinitely many x € [0,1], & f is not continuous at infinitely many x € [0,1]?
41	AR,CK	Is there a function defined on [0,1] and continuous no where such that if x € [0,1] the function is not continuous at x?
42a	CK	Suppose f:A --> B is 1-1 & onto (i.e. bijective, note: f inverse: B --> A exists & is bijective) & a � A, then f inverse is continuous at f(a).
42b		Suppose f:A --> B is 1-1 & onto & bijective (i.e. bijective, note: f inverse: B --> A exists & is bijective) & a � A, then f inverse is continuous at f(a).
43		If f:[0,1] --> R then there exists x � [0,1] such that f (x) >= f(y) for all y € [0,1].
44	AR,VM	Find a continuous, bijective f:(0,1) --> R.
45a		If f:[0,1] --> [0,1] is continuous & f(0) = 0, f(1) = 1 & y � [0,1], then there exists x � [0,1] such that f(x)=y.
45b	CK	If f:[0,1] --> [0,1] is continuous & f(0) = 0, f(1) = 1, then there exists x � [0,1] such that f(x)=x.
45c		If f:[0,1] --> [0,1] is continuous, then there exists x � [0,1] such that f(x)=x.
46	(b)HH	Find the limit point(s) of the following sets. (These can be done separately.) A = {n * 2^(1/2) mod 1: n is an element of Z+} (a mod 1 is the decimal portion of a). B={1+1/2+1/3+...+1/n: n is a positive integer} C={e(1)1+e(2)1/2+.....+e(n)*1/n: n is a positive integer and e(i) is in {1,-1}for each i}. D = Let a be a number, f be the real valued function defined by f(x) = x2 - 2, and define Sa = {fn(a):n is a positive integer}. Here f1(x) = f(x) and fn(x) = f(fn-1(x)).
47		If f:A --> B is continuous and s is an open/closed subset of B, then the pre-image of S = {a � A: f(a) � S} is open/closed.
48	AR	If A is countable, then A has length 0.
49	VM, CK	If T is a sequence with limit 0, p is a number, and S is a sequence defined by S(n)=T(n) + p, then S has limit p.
50	HH	If A is a number set and p is a limit point of A, then there is a sequence S such that S(n) is in A for each n and S has limit p.
51	CK	The sequence S has limit p iff each subsequence T has limit p.
52	RJ, HH	If the sequence S is bounded, then S has a subsequence T that has a limit.
53		If S is a sequence which satisfies if c>0 there is a positive integer N such that if i,j > N then \|S(i) - S(j)\|<c, then S has a limit.
54	TO(c)	Suppose D is a subset of R, f is a real valued function with domain D, and p is in D. The following statements are equivalent: a. f is continuous. b. if p is in D and c>0 then there is a d>0 such that if x is in D and \|x-p\|<d then \|f(x)-f(p)\|<c. c. if U is an open set, then f-1(U) is open relative to D. d. if K is a closed set, then f-1(K) is closed relative to D. e. if p is in D and S is a sequence in D with limit p, then the sequence T defined by T(n)=f(S(n)) has limit f(p).
55	CK	If f is continuous over U and V is contained in U, then f restricted to V is continuous.
55b		Suppose f:D --> R, where D is a countable dense subset of [0,1] and it is continuous over D, then there exists a function g:[0,1] --> R, such that G restricted to D ='s f
55c		If D is a dense subset of [0,1] then D contains a countable dense subset of [0,1].
56	AR	If we denote the Cantor set as set J, show J is uncountable.
57	AR	Is 1/4 � J?
58	AR	If w is countable and c > 0 there exists a sequence of invervals such that if x �W then x � (a sub n, b sub n) for some n � Z+ and (summation sign sub i) b sub i - a sub i < c .(i.e. if w is countable then w has length 0)
59	AR	Find the length of J.
59b	AR,CK	Find the length of J. Without using the compliment equals 1.
60		Is there a f: [0,1] --> s.t. f is continuous at each irrational, but discontinuous at each rational?
--	-------	----------------------------------------------------------------------------------------------
###	NO ONE!	5b,28,31b,38c,42b, 43, 45a, 45c, 47, 51 - 53,55b,55c, 60

AR	Abraham	64	1
AH	Amy
CK	Crystal	51	1
CT	Candy	66a	1
DG	Dasd;flksdf
HH	Haku	52	1
MG	Manard
RJ	Ruth	52	1
TA	Tiffy
TO	Tina
VM	Vanessa
WW	Wendy

Math 431

Nickname

Problems:

#

Finished by

Scary Problems

T

HH,VM

P

AH

1

VM,CK

2

DG

3

AR

4

VM, RJ, TO

5

RJ, MG, AR

5a

DG

5b

DG

6

CT, VM

7

CT

8

DG

9

VM,

10

AR

11

WW

12

CT,RJ

12b

VM,

13

DG

14

HH,AR

15

VM, TO

16

DG

17

CK

18

AR, TO, DG

19

TO

20

WW,VM

21a

RJ

21b

VM

22a

CK, VM

22b

CK,VM,RJ,DG

23

CK

24a

MG, VM,AR

24b

RJ

24c

HH,VM

25

AR,DG

25b

CK, VM,DG

26

CK, VM

27

WW, VM, AR