Midterm 2

Question 1

What are the Axioms of Natural Numbers?

Answer 1

For s on N: One to One, Onto, s(x) = s(y), Only one “1” in N, and it is not the successor of any number, If given T, which contains 1 and s(x) is in T, then T = N.

Question 2

True of False? A = {A} for object A?

Answer 2

False. {A} is the set. A is the object

Question 3

Given a Subset A, what can we assume about A?

Answer 3

A is a subset of itself. {0} is a subset. Two Sets are identical if they have the same members . All A is in B and All B is in A. Subsets of A is marked as 2^A. B is in 2^A if and only if B is in A.

Question 4

Define a Subset

Answer 4

For each object x in A. It is true that x is in B. Thus A is a subset of B. A ‘C’ B.

Question 5

A Symbol - ‘C’ - B

Answer 5

A is contained in B

Question 6

T/F (a,b) ‘C’ [a,b] ‘C’ R

Answer 6

True. Because the open interval is a subset of the closed interval

Question 7

A ‘E’ 2^A

Answer 7

True, Because A ‘C’ A

Question 8

A ‘C’ (is a subset of) 2^A

Answer 8

False, Because each x in A must be in 2^A and an element of a is not a subset of A.
{A} ‘C’ 2^A is True

Question 9

{O} ‘E’ 2^A

Answer 9

True, {0} is a subset of A.

Question 10

{0} ‘C’ (is a subset of) 2^A

Answer 10

True, {0} is a subset of A

Question 11

There are no members in {0}

Answer 11

False, 0 is a memeber

Question 12

Intersection vs. Union

Answer 12

n vs U and intersections is the elements they share, while Union is elements in either A or B for A U B.

Question 13

Complement of S, A

Answer 13

All Elements in S that are not in A. S/A or S - A

Question 14

Demorgan’s Laws

Answer 14

c(A U B) = c(A) n c(B)
c(A n B) = c(A) U c(B)
c = compliment

Question 15

A is a subset of B iff, A U B = B

Answer 15

True. All A are in B, thus A subset of B

Question 16

A is a subset of B iff, A n B = A

Answer 16

True, The intersection of A,B being A means the two sets share all A. Thus A is a subset of B.

Question 17

A is a subset of C(B) (compliment of B), iff A n B = {0}

Answer 17

True, So A and B share no elements between them, thus A must be in the elements that are not in B.

Question 18

Compliment C(A) is a subset of B iff, A U B = S. We know A is a subset of S and B is a subset of S. S is universe

Answer 18

If x is not in A, then x is in B and since the A U B is the universe it must be in B. Therefore True

Question 19

A Subset of B iff C(B) Subset C(A)

Answer 19

True, Because if all elements not in B are elements not in A, then the elements in complements of each of these complements is the same. use the C(C(A)) = A fact.

Question 20

Bijection, Injection, Surjection

Answer 20

Bijection = One-to-One & Onto, Surjection = Onto, Injection = One-to-One.

Question 21

Define a Metric Space

Answer 21

Metric Space is a Set of numbers and a prescribed quantitative measure of the degree of closeness of pairs of points in this space. A set of numbers with a distance function.

Question 22

4 Properties of Metric Spaces

Answer 22

1. Distance >= 0
2. Distance(x,y) = 0 iff x = y
3. Distance(x,y) = Distance(y,x)
4. Distance(x,y) <= Distance(x,z) + Distance(y,z)

Question 23

What is Compactness?

Answer 23

If every sequence in S has a subsequence that converges to an element again contained in S. A Set is compact only if it is closed and bounded.

Question 24

What is the Bolzano-Weierstrass theorem?

Answer 24

Every bounded sequence of Real Numbers has a convergent subsequence. If X is a closed Bounded subset of R, then every sequence in X has a subsequence that converges to a point in X.

Question 25

How is the Bolzano-Weierstrass Theorem used?

Answer 25

It states every continuous function on a closed bounded interval is bounded.

Question 26

What is a Cauchy Sequence?

Answer 26

Say you have a sequence of Real Numbers {a}. Then a is a Cauchy sequence if for every e > 0, there is a Natural number N so that when n, m > N we can say |an - am| <= e.

Question 27

What is a Cover? Open Cover?

Answer 27

A cover C of X is a collection of subsets of X whose Union creates the whole space of X. Thus if Y is a subset of X, then a cover of Y is a collection of subsets whose Union is Y.
Open Cover is a cover when the elements are all open sets.

Question 28

What Is a Subcover?

Answer 28

A subcover is just a subset of the cover such that this subset still covers the entire space.
Let C be a cover of a topological space X. A subcover of C is a subset of C that still covers X

Question 29

Can you map R onto [0, 1]?

Answer 29

Yes, The Function: f(x) = 1 / (1 + e ^-x) used in machine learning and neural networks.

Question 30

What is density in metric spaces?

Answer 30

A Subset D of a topological space X is Dense in A if A is in D. Thus a Set D is dense if and only if there is some point of D in each non-empty set of X.
a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A
Every topological space is a dense subset of itself.
Nowhere dense means empty or if the closure of A has an empty interior.
s = Serbs, B = Bosnians. It is nowhere dense if we can make an open gate around any Bosnian and not have nay serbs within it. It is dense if we cannot make any open fences around a Bosnian that has no serbs in it. Serbs contain the Bosnians to isolated points

Question 31

What is a countable set?

Answer 31

A set S is countable if there exists an injective function f from S to the natural numbers N = {0, 1, 2, 3, …}
the elements of a countable set can always be counted one at a time and, although the counting may never finish, every element of the set is associated with a natural number

Question 32

Examples of Density?
Q & R
Z & R

Answer 32

Q is dense in R, because its limit points are all real numbers and its closure gives R.
Z is nowhere dense in R because Z has no limit points and hence its closure is itself. Z does not contain any open intervals. Z is all isolated points, there is no need for limit points.

Question 33

Explain Nowhere Dense Set?

Answer 33

Set whose closure is an empty set. Any subset of nowhere dense is nowhere dense. Union of finitely many nowhere dense sets is nowhere dense. Union of countably infinite nowhere dense sets might not be nowhere dense. Complement of closed nowhere dense set is the open dense set.
A nowhere dense set is always dense in itself.
{Z} is nowhere dense in R
Put another (equivalent) way, a set A is nowhere dense iff for every non-empty open set U there is a non-empty open set V such that V⊆U and A ∩ V=∅.

Question 34

What is an interior?

Answer 34

An interior of S is all the points of S that do not belong on the boundary of S. The interior is the compliment of the closure of the compliment of S.
Another way to write it is:
for s in E is interior to E if for some r > 0 we have { d(s, s0) < r} in E.

Question 35

What is connected?

Answer 35

When it impossible to write x as the union of two disjoint open sets. When a plane has an infinite line through it then it is disconnected.

Question 36

Why are the rationals dense in R?

Answer 36

Between any two real numbers there is a rational. The integers are not dense in R because we can find 2 reals with no integers between them.

Question 37

Difference Between LimSup and Lim

Answer 37

Limit can exist for partial sequences in {a}, but Lim Sup is one value for the sequence. The Lim Sup can exist and the Lim may not. Ex. {3, -7, 7, -1, 1, -1, 1…..
The limit does not exist, but the Lim Sup = 7.

Question 38

Say Lim Sup(SnTn) where Sn converges to S. What can we say about the Lim?

Answer 38

We can change it to

Lim Sup Sn Tn = s * limSup(Tn)

Question 39

For sequence Tn & Sn. What does it mean for lim Sup(Tn + SN) and Lim Sup(Tn * Sn)?

Answer 39

Tn + Sn means you have all elements of Tn added to all elements of Sn. Like add index 0 of Tn with index 0 of Sn and only with matching indices. Tn * Sn is the same way, but you multiply it.

Question 40

Prove Lim inf(Sn + Tn) ?? Lim inf(Sn) + Lim inf(Tn)

Answer 40

>=. And the proof is that for some e > 0.
 t - e/2 < tk for k > N1. 
And s - e/2 < sk for k > N2
Then for n > Max(N1, N2). 
(s - e/2) + (t - e/2) < (tk + sk)
s + t - e <= Lim {inf(tk + sk)}
Lemma: If x - e <= w for all e > 0, then x <= w.
Thus Proven True

Question 41

Balzano Weierstrass Theorem

Answer 41

Every Bounded Sequence in R has a convergent subsequence.

Question 42

Difference Between Max & Supremum of a Set

Answer 42

If a Set has a Max, Then MAx = Supremum. Max may not exist, but Supremum does. Max (a, b) doesn’t exist, but Supremum (a,b) = b. Same for Infinimum.

Question 43

Axiom of Completeness

Answer 43

Every Nonempty Subset of R that is bounded above has a least upper bound. Sup (S) exists and is a real number.

Question 44

How do we tell if a Metric space (S, d) converges? How does it prove a Cauchy Sequence?

Answer 44

Metric Space (S,d) converges to s in S if the Lim n -> Infinity of d(sn, s) = 0. This can also be used to prove it is a Cauchy Sequence. e > 0 where m,n > N then d (sm, sn) < e.

Question 45

When is a Metric Space Complete?

Answer 45

If every Cauchy Sequence in S converges to some element in S.

Question 46

What does E^o signify?

Answer 46

Set of points in the Interior of E. The Interior of E is a point strictly within the boundary.

Question 47

What does it mean for a subset E of a Metric Space S to be closed?

Answer 47

Then its compliment S \ E is an open set.
The intersection of any closed set is closed.
A set is closed if E = E^-
A set is closed if it contains the Limit of every convergent sequence of points in E.

Question 48

What does it mean for Subset E of metric space S to be open?

Answer 48

Then E is open in S if every point in E is interior to T. E = E^o. S is open in S.
Empty Set is Open in S.
Union of any collection of open sets is open
Intersection of finitely many open sets is again open set.

Question 49

What is the Closure of E? Or E^-? Subset E of a Metric Space S.

Answer 49

The intersection of all closed sets containing E.

An element is in the Closure of E if it is the limit of some sequence of points in E.

Question 50

What is the boundary of E? Subset E of a Metric Space S

Answer 50

It is the Set Closure(E) / Interior(E).
A point is in the boundary of E if and only if it belongs to the closure of E and its compliment.
Always Closed

Question 51

Difference between an open and closed set. Example of Open Set, Closed Set, Neither, Both.

Answer 51

A set is considered open if we can draw a ball around any point within the set and have that point be contained in the set. A set is closed if its compliment is the open set. Closed and Open are NOT Opposites. If a set is not open then it does not mean it is closed.

(0,1) is open because we can draw a ball around any point in (0,1) and see that there is some interval that contains that point aka some ball.

[0,1] is closed because we can see that the compliment of [0,1] - namely (-infinity, 0) U (1, infinity) is open.

Neither is (0, 1]

Both is Empty Set and R

Question 52

What is the interior of [a,b]?

Answer 52

(a, b)

Question 53

What is the boundary of both (a,b) and [a,b]?

Answer 53

The two element set {a, b}

Question 54

What is Compactness?

Answer 54

Heine-Borel Theorem States that A subset of Rk is compact if and only if it is closed and bounded.

Question 55

What is the closure/Interior of for these sets in R?

a. {1/n : n in Naturals}
b. Q (rationals)
c. Cantor Set
d. { r^2 < 2 for r in Q}

Answer 55

a. Interior = 1/n for n in naturals. Closure = {1/n n in naturals} U {0}.
b. Interior = Empty Set. Closure = R
c. Interior = Empty Set, Closure = Undefined
d. Interior is r^2 < 2. Closure = [-root(2), root(2)]

Question 56

What is an Interior of a set S?

Answer 56

A set strictly within the bounds of S and not on the boundary.
A point q is an interior point of E if there exists a ball at q such that the ball is contained in E. The the interior set is the set of all interior points. But it really depends on what the set is a subset of. If Q is a set of Q, then the interior is Q, but if Q is a subset of R, then the Interior is empty Set since we can’t take any Q and create an interval with no reals in that interval.

Question 57

What is the closure of a set S?

Answer 57

The set containing the boundaries and all the limits of the set S. Thus it will usually include the elements in the set. as well.

Question 58

How to tell if an infinite summation ∑ converges?

Answer 58

Prove that the sequence of partial sums converges to a real number. Find the limit of the sequence and see if it converges to some Real s. ∑ A converges if Lim A = 0. Breaking up into the subsequences also implies that a Series converges only if it satisfies Cauchy Criterion.

Question 59

What does it mean for ∑{A} to converge absolutely?

Answer 59

If ∑|A| converges then converges absolutely. If Converges absolutely then it is convergent.

Question 60

Geometric Series?

Answer 60

∑ar^k = a(1 - r ^(n + 1)) / (1 - r). If you want to prove this, then make sure if starts at 0 and goes to infinity and it has an ‘a’ and an ‘r’ setup just like this.

If r < 1, then
∑ar^k = a / (1 - r)

Question 61

P Rule?

Answer 61

∑1/r^p converges if and only if p > 1.

Question 62

Explain a Cauchy sequence Sn?

Answer 62

Sn is a Cauchy sequence if for some e > 0 there exist an N such that n, m > N and |Sn - Sm| < e.
Every converging sequence is a Cauchy Sequence.
Every Cauchy Sequence Converges.

Can you find N so that any pair of elements after N
is within a distance of 1 from each other?

Question 63

What is the Ratio Test?

Answer 63

A Series ∑An of NON-ZERO terms
converges absolutely if lim sup(|A(n + 1) / An|) < 1
diverges if Lim Inf(|A(n + 1) / An| > 1
Undetermined if = 1.

Question 64

Root Test

Answer 64

For Series ∑A . Let a = Lim Sup |A|^(1 / n).
Converges if a < 1
Diverges if a > 1
Undetermined if a = 1.

Question 65

Comparison Test

Answer 65

∑A is a series where An >= 0 for all n.
If ∑A converges and Bn <= An for all n, then ∑B converges
If ∑A = infinity and Bn >= An for all n, then ∑B = infinity.

Question 66

Consider ∑ 1/ (1 + n^2)

Answer 66

compare to 1 /n^2 to show it converges. p test.

Question 67

Prove that ∑1/n diverges

Answer 67

Use Integral. 1/x from 1 to n + 1 is ln(n+ 1) which is infinity.
Same proof for 1/x^2, but you don’t have an ln.

Question 68

Integral Test

Answer 68

Ratio and Root and Comparison do not apply.
Terms in ∑A are nonnegative
There is a decreasing function f on [1, infinity] such that f(n) = An for all n.
To be decreasing, if x < y, then f(x) < f(y)
If lim of the [integral from 1 to n] f(x)dx = infinity then diverges if < infinity than converges.
This allows us to proves that series like ∑(-1) ^n / root(n) converges, but does not absolutely converge.

Question 69

Alternating Series Theorem

Answer 69

If a1 >= a2 >= a3 >= a4 >= a5 >= an >= 0 and lim an = 0, then the alternating series converges.

Question 70

Decimal Expansion Rules

Answer 70

1. Every nonnegative number has at least one decimal expansion.
2. A Real number x has exactly one decimal or two (one ending in all 9s and one ending in all 0’s).
3. A number is Rational only if its expansion is repeating.

Question 71

Write the Fractions as Decimals:

0. 2
1. (2 repeating)
2. (02 repeating)
3. (14 repeating)
4. (10 repeating)
5. 1(492 repeating)

Answer 71

0.2:
10x = 2.0 (multiply to get repeating part to the left of decimal point)
Repeating already to the Right.
10x = 2.0 and thus x = 1/5

0.(2 repeating):
10x = 2.(2 repeating) (Repeating on Left)
x = 0.(2repeating) repeating Right
Subtract two equations:
9x = 2. 
x = 2/9!

0.1(492 repeating)
10000x = 1492.(1492 repeating)
10x = 1.(492 repeating)
Thus 9990x = 1491
x = 1491/9990

Question 72

Convert the Fractions to Decimals

Answer 72

Just Do long Division or where possible try to multiply to get a power of 10 on the bottom and then you can get the decimal really easily.

Question 73

What does it mean for function f(x) to be continuous?

Answer 73

f(x) is continuous at x0 in dom(f) if, for every sequence (xn) in dom(f) converging to x0, we have lim f(xn) = f(x0). Then f is continuous on S. S in dom(f).
|f| and kf must be continuous as well.

Question 74

Prove that f(x) = 2x^2 + 1 is continuous for x in R.

Answer 74

Suppose Lim Xn = X0. Then we have Lim f(Xn) = Lim [2(Xn)^2 + 1] = 2[LimXn^2] + 1 = 2(X0)^2 + 1 = f(X0).

Question 75

Composite Theorems of continuity?

Answer 75

If f is continuous at x0 and g is continuous at f(x0), then composite g o f is continuous.

Question 76

What is the Domain of √(4 - x)?

Answer 76

(-∞, 4]

Question 77

Closed Interval Function

Answer 77

Then the function is bounded and f assumes its maximum and minimum values on [a,b]. Exists x0, y0 in [a,b] such that f(x0) ≤ f(x) ≤ f(y0)

Question 78

Continuity vs Uniform Continuity

Answer 78

Uniform continuity is for a set, continuity is for a point or interval. Something is Uniformly continuous for a function or a set. It doesn’t make sense that a function is uniformly continuous at each point.

Question 79

Show f(x) = x^2 is uniformly continuous on [-7, 7].

Answer 79

e > 0. |f(x) - f(y)| = |x^2 - y^2| = |x - y||x + y|.
|x + y| ≤ 14. Then |f(x) - f(y)| ≤ 14|x - y|. ∂ = e/14.
Then |x - y| < ∂ implies |f(x) - f(y)| < e.

Question 80

If F is continuous on [a,b], then?

Answer 80

it is uniformly continuous on [a,b].

Question 81

Show f(x) = 1/x^2 is not uniformly continuous on (0,1).

Answer 81

Sn = (1/n) and then (Sn) is not a Cauchy Sequence. Thus f(Sn) cannot be uniformly continuous on (0,1)

Question 82

A function f on (a,b) is uniformly continuous on (a,b) if ?

Answer 82

it can be extended to a continuous function f´on (a,b).

Question 83

How can i prove a function is continuous?

Answer 83

If Lim x -> a f(x) = f(a), for all a in dom(f(x)) then it is continuous.

Question 84

How can i decide if a function is continuous on [a,b).

Answer 84

use the fact that a function is continuous on (a,b) if it is continuous on [a,b].

Question 85

How do we prove a function on one metric space S to S* is continuous at s0? f: (S, d) -> (S, d).

Answer 85

For each e > 0. there exists ∂ > 0 such that d(s, sO) < ∂ implies d(f(s), f(s0)) < e.
A function f : S -> S is continuous if and only if f^-1(U) is an open subset of S for every open subset U of S*.

Question 86

For metric spaces (S, d) and (S, d) and a continuous function f: S -> S*. Let E be a compact subset of S. Then?

Answer 86

f(E) is a compact subset.
f is uniformly continuous on E.
f is bounded on E.
f assumes its maximum and minimum on E.

Question 87

What is countable?

Answer 87

Anything that can be listed as a sequence.

Question 88

Properties of Metric Space (S, d)?

Answer 88

a. If Un is a sequence of dense open subsets in S, then the intersection of these sets is dense in S.
b. If F is a sequence one closed subsets of S and the union of the F’s contains a non empty open set, then so does at least one of the sets of F.
c. Unions of sequence of nowhere dense subsets in S has a dense compliment.
d. Space S is not a union of a sequence of nowhere dense subsets of S.

Question 89

What does disconnectedness entitle?

Answer 89

A space which is a union of two disjoint non-empty open sets is called disconnected.
Take Open Sub Sets U1 and U2. n = intersection.
(E n U1) n (E n U2) = ø
and E = (E n U1) U (E n U2)
when (E n U1) ≠ ø and (E n U2) ≠ ø.
Or
If E = A U B and (closure A) n B = ø and A n (closure B) = ø.

Question 90

First Category vs. Second Category

Answer 90

Sets are either First or Second Category. S is second category if it cannot be written as the countable union of subsets which are nowhere dense in S. First Category are not second category.
Rationals in R are First Category
Irrationals in R are Second Category
Z inherited from subset of R is Second Category since every subset of Z is open in Z.
Z in R is First category.