5

Letand M be On-empty sets. For each a € A; let Ma = M_ Let U be an ultrafilter On A, Let II Ma /~be as defined in the Structures notes_ Show that A < II Ma...

Question

Letand M be On-empty sets. For each a € A; let Ma = M_ Let U be an ultrafilter On A, Let II Ma /~be as defined in the Structures notes_ Show that A < II Ma aEA aeA by defining an injection g : A - II Ma and proving that g is an injection_ aeA Let everything be as in the previous question and further assume that M is finite. Prove that there is & bijection 9 : A - II Ma aEA

Let and M be On-empty sets. For each a € A; let Ma = M_ Let U be an ultrafilter On A, Let II Ma /~be as defined in the Structures notes_ Show that A < II Ma aEA aeA by defining an injection g : A - II Ma and proving that g is an injection_ aeA Let everything be as in the previous question and further assume that M is finite. Prove that there is & bijection 9 : A - II Ma aEA



Answers

Suppose S is the smallest ?-algebra on R containing {(r, r + 1) : r ? Q}. Prove that S is the collection of Borel subsets of R.

Right. So I want to show that the integers are equivalent to the natural numbers. So we construct our function. So it takes the integers to national numbers. And we will define it by have a fantasy call. So Piecewise function. So to end plus one. Okay, negative to it. This is where none is better than zero when it is strictly a sincere. Oh okay. So our natural numbers they can be positive or negative but natural numbers by definition of like positive. Right? Let's start with one to keep going. So our god and um even integers they have to be positive. And so this is even but we put negative in there to make sure that you didn't positive. Um National number. Okay. So we want to see if this is by objective. So um Is it injected 1st? Yeah. Um we let CN one and 2 be in uh however positive integer. So non negative integers. And then right. And seeing such that F of and one is equal to half of and to you then have Okay, two and one last one is equal to To end two plus one. This gives us and want to see also. And so Okay. And next we consider the possibility when both are negative introduce so such that our outcomes are the same outfits are the same. Then we have negative and one is equal to negative two and this gives us That and one is equal to. And so and finally we'll consider when pair of vintages we choose have different science. So but then one B in uh huh the non negative integers. And uh and to be in the negative images. So should that well we can't use the same possible here. So we have um to difference integers that have different signs. We will use the definition of objective function that says that if I have so and one is not equal to into one is positive and the other one is negative. So if I have two different imports I should get to different apples. Thank you then. Yeah. Hi f If anyone support to two and one plus one that's positive, not negative is equal to was not equal to -2 and two I mean which is equal to have offensive. So I have two different inputs. I have two different output. So this is injected. So it proved like three cases of interactivity when we have both of our interests positive negative. We both have different um science so it's objective now for such activity of the so that uh I'm not sure number that's even Mhm. And then you choose mhm an integer and as negative over to. So this is our even natural numbers and I want to find and just move this around get them -2 of a negative too and know that this is an integer. Yes. Mhm. Then okay. If a fan is equal to to let you sequel to see to native to times negative came over to and this is the culture. Yeah. Now that's um the odd, it's the same concept we choose and which gives us And -1/2. Okay. And this is an instant er then F. N. Is equal to F off And that's one of the two to see baltar two times and -1/2. That's from this gives us a minus one plus one, which is so function if this objective and has the interviewers equivalent to the natural numbers.

In this problem, we are given to subsets of our name, the S. And T. And we have the property that given us and T. S. Is going to be less than or equal to T. For every S. And S. And T. N. T. So we're first supposed to show that S. Is bounded from above. So in order to do that, we just pick a tea from tea. So it's like T. One B. From T. Then for every S. And S. S. Is less than or equal to D. One us founded above. Similarly for bounded below. For S. Let her T. Sorry, let S. One B. And S. Then for every T. And T. S. One is less than or equal to T. Us founded. Hello. We're now supposed to show approve that's the supreme um of S. Is less than the infamous um of T. So let's do this by contradiction. So suppose this is not the case. That's a pretty mom of S strictly greater than inform um of T. And this implies there exists some S. In S. Such that the S. Is greater than the infamous um of T. Furthermore there exists a T. Into such that the in form um of T. Is less than or equal to T. It's less than S. Because otherwise if that wasn't the case S. Would be the in form. Um Right. And we have picked that S. Is greater than the infamous. Um Yeah so that means there exists T. And S. Such. That T. Is less than S. But this is a contradiction because all asses are less than three. Quality us. The in form um O. T. Is less than or equal to the supreme um of us. So then we're supposed to come up with a couple of examples where we have S. Intersect T. Is not the empty set. And we can simply put this as well. S. B. 0 to 1 and that T be 1 to 2. All right. Every element of S. Is less than a record of every element of T. And the intersection it's not the empty set. We want. Now an example where the supreme um of S. Equals the in form um of tea but we want s intersex T. To be the empty set. So how can we do this? Well we can take our other example and just open it up. Sp 0 to 1. The open interval and let T. V. One two. The closed interval open interval. So the in form um of T. Is one, the supreme um of S. Is one and those are equal.

Zero is less than B is less than a. Let's consider the function A four X. Is equal to expire in and is it and teacher great and equal to do now there is a theory called mean value to you. This function is a polynomial. It's continuous and differentiable. In an interval beat way. It's continuous enclosing to Albert way and differential. An opening to a gateway. Right? Very function is continuous in a closed into a gateway and differential and opening to a gateway then like green just mean value to them like ranges mentality and says that F a B minus ffb divided by a minus B is equal to a dash of something between be any. A very powerful freedom in calculus. Lagrange is humanitarian. Let's use this. So what is my fook? Fook will be a parent minus B. Baron divided by a minus B. Is equal to a dash of. See what is the derivative of this function? It is an ex power in minus one. This is the radio off function. Experiment by power. So what will be a flash of C. N. C. Bar and minus one? All right now you can see here sees between B. And so that means he's less than me. You see is less than a C. Bar and minus one is less than a bar and minus one. Multiplying and pull tight N. C. Bar and minus one is less than in a bar in minus one. So this quantity is less than in a par and minus one. So that would imply a par n minus B part and divided by a minus B is less than any power and minus one for all and greater than or equal to two for one. What will happen is it will be exactly equal because a pas one minus people. One divided by a minus B. It's compared with one paper, one minus one. This is one and 180 is one. So for one when it is one, this particular inequality is actually an equal. But when you increase one, that is when you take higher than 12345 all natural numbers starting with two. This inequality will hold two and we included by Lagrange is mean value.

Okay. So the question is as follows, where we look at the status, which we say is the smallest sigma algebra, What is a signal of zero generated by the sigma algebra in in our right, generated by the set are Come on R Plus one. Where are is a rational number. So this is the problem statement should be on our Okay, so moreover, we need to show that S is the collection of moral subsets of our So we need to say that S is the borough subsets. Okay, well the barrel uh, the barrels sat on our is you can think of it as the smallest. Take my algebra generated by uh let's say a basis for the topology of our and a basis for the topology of our is just as I said, A B where am br elements of the reels. Okay, so the topology, the standard topology and our is generated by such sets. And if we can show that S contains uh a set of the form, A comma, B where A and B are real numbers, then that's sufficient to demonstrate. That. Is is uh wow the moral sigma algebra on our and the way that we do this is what we grab one of these. Uh the goal here is to make that said a B. Right. Where A and B are real numbers. You're seeing or from Sets of the form are comma are plus one where are is irrational. Uh taking these sets to these such sets to be an element of sigma algebra. Okay then the way that we do this is let me make some space for these. Oops. So let's look at this set A Excuse me? A comma eight plus one. Right. And uh let me let me do something. A sub N. Coma A sub N plus one. Where S A Ben is a series of rational numbers, right? Such that he saw Ben converges to a rational number. A uh sorry, a real number A From the right. Okay. Then if we take the union over end of all. A seven come on in some men Plus one, guess what? We're going to get this set A comma a Plus one. And this set will be an element of S. And it's important to note this because A Now is a real number. Right? And we want to do something similar. Right? Now we look at the set B. Seven come up beasts up and plus one where of course be summoned once again is rational, right? And we say that Visa ban converges to be from the left and B is a real number. Then it turns out that the complement of this set complement of this set, which will be let's look at let me copy this. I don't have to rewrite it. Once again, the complement of this set will be the same from negative infinity coma be suburban Union be. So then plus one comma infinity. Right? And this set will be in S because see my algebra. Czar close with respect to compliments. And now if we grab this set and we intersected with the set that we found down here, so let me grab it and copy they grabbed the set right, Win or sickbed. We'd are set here. It's what we're going to get. We're making the assumption. Obviously that is less than strictly less than the So keep that in mind. Well, if we do these, we're going to get this set a comma B. So then closed. And if we take the union over end of these sets, we will get a comma B, which is the goal that we were trying to achieve. So the sigma algebra, in conclusion, the sigma algebra generated by this set contains sets of this, the following form and all sets of the following form. And we know that all such sets for my basis for the topology of our therefore the city. S. Is the world sigma algebra owner


Similar Solved Questions

5 answers
Problem 10.39Consider the shaded area shown in (Figure 1) : Suppose that 8 in, b =Part ADetermine the moment of inertia of the shaded area about the y axis_ Express your answer to three significant figuros and include the appropriate units_inSubmitPrevious Answcrg Request AnswerIncorrect; Try Again; attempts remainingProvide Feedback1 of 1Figure
Problem 10.39 Consider the shaded area shown in (Figure 1) : Suppose that 8 in, b = Part A Determine the moment of inertia of the shaded area about the y axis_ Express your answer to three significant figuros and include the appropriate units_ in Submit Previous Answcrg Request Answer Incorrect; Try...
5 answers
Use the References access important ralues if needed for this question:In the laboratory student combines 26.2 mL of a 0.258 M sodium hydroxide solution with 13.0 mL ofa 0.690 M barium hydroride solution_What is the final concentration of hydroride anion818 Submit AnswerRetry Entire Groupmore group attempts remaining
Use the References access important ralues if needed for this question: In the laboratory student combines 26.2 mL of a 0.258 M sodium hydroxide solution with 13.0 mL ofa 0.690 M barium hydroride solution_ What is the final concentration of hydroride anion 818 Submit Answer Retry Entire Group more ...
5 answers
Of 5 MHz and wavelength of 300um Anultrasound pulse having a frequency: What is the speed = of sound in that material? (micrometer) passes through a material6000 m/s150 m/s60 mls1500 m/s
of 5 MHz and wavelength of 300um Anultrasound pulse having a frequency: What is the speed = of sound in that material? (micrometer) passes through a material 6000 m/s 150 m/s 60 mls 1500 m/s...
5 answers
COzEtCOzEtH3OtCOztBuOH
COzEt COzEt H3Ot COz tBuOH...
5 answers
QUESTION 8A lower P-value resulring from a null hypothesis significance rest; generally means_ More knowedgeable researchers conducted the statistical analysis stronger evidence against the null hypolhesis and support for the alternative hypothesis; More evidence Jgainst lhe alternalive hypothesis Astronger Case tor increasing the statisucal power of Ihe lesl
QUESTION 8 A lower P-value resulring from a null hypothesis significance rest; generally means_ More knowedgeable researchers conducted the statistical analysis stronger evidence against the null hypolhesis and support for the alternative hypothesis; More evidence Jgainst lhe alternalive hypothesis ...
5 answers
F(x) Given the functionx2 + 3(a) Find the domain.(b) Find the intercepts.(c) Find the asymptotes. Showyour work clearly using limits (d) Find the intervals of increase or decrease. Find any local minimum and maximum points(e) Find the intervals of concavity: Find any inflection point(s)_Use the information from parts (a) (e) to sketch the graph: Label clearly any axes,intercepts, asymptotes
f(x) Given the function x2 + 3 (a) Find the domain. (b) Find the intercepts. (c) Find the asymptotes. Showyour work clearly using limits (d) Find the intervals of increase or decrease. Find any local minimum and maximum points (e) Find the intervals of concavity: Find any inflection point(s)_ Use t...
5 answers
Find $w(x), a$, and $b$ for the case of the classical orthogonal polynomials in which $s(x)$ is of second degree.
Find $w(x), a$, and $b$ for the case of the classical orthogonal polynomials in which $s(x)$ is of second degree....
5 answers
Consider the power seriesZax(x c)zk k=0 where c=2and given thatantl Iim n - anwhere L=27What is the left side endpoint of the interval of convergence? Round to the nearest hundredth(For example; if the interval of convergence is (-4,6], then the left side endpoint would be -4.)
Consider the power series Zax(x c)zk k=0 where c=2 and given that antl Iim n - an where L=27 What is the left side endpoint of the interval of convergence? Round to the nearest hundredth (For example; if the interval of convergence is (-4,6], then the left side endpoint would be -4.)...
5 answers
ClullelnStattus: anFaStkh of the followlng is the major product of the reaction below?(CH,KCOocichy}0 3 O 0
clullelnStattus: anFa Stkh of the followlng is the major product of the reaction below? (CH,KCO ocichy} 0 3 O 0...
1 answers
Complete each statement about $\square K M P R$. Justify your answer. $$\angle M P R \cong \underline{?}$$
Complete each statement about $\square K M P R$. Justify your answer. $$\angle M P R \cong \underline{?}$$...
5 answers
The linear density rod m long is 8/Vx + 9 kg/m, where x is measured in meters from one end of the rod. Find the average density Pave of the rod. Pave kg/m
The linear density rod m long is 8/Vx + 9 kg/m, where x is measured in meters from one end of the rod. Find the average density Pave of the rod. Pave kg/m...
5 answers
101.402 The following objects are released at the tOp of 4 simultaneously from rest eS0-m-long Tamp inclined at 3.508t she horizont tal: solid sphere , solid cylinder; a hollow cylindrical hollow ball: (a) Which wins the shell, and winner reaches the race? (b) At the moment the objects. boltom, find the positions of the other three 102 80 solid .
101.402 The following objects are released at the tOp of 4 simultaneously from rest eS0-m-long Tamp inclined at 3.508t she horizont tal: solid sphere , solid cylinder; a hollow cylindrical hollow ball: (a) Which wins the shell, and winner reaches the race? (b) At the moment the objects. boltom, find...
5 answers
For each of the following, find the next four terms of the sequence, assuming n=l. 3* Cnbn (-1)2n+1an = n + (n + 1) + (n + 2) + :+ Znb1 =2, bz = 3,bn Zbn-1 + bn-2
For each of the following, find the next four terms of the sequence, assuming n=l. 3* Cn bn (-1)2n+1 an = n + (n + 1) + (n + 2) + :+ Zn b1 =2, bz = 3,bn Zbn-1 + bn-2...
5 answers
1-T T' T (-)"n?" +~+ 2! 41 6! (2n)!A)B) 0C) -12D)E) T
1-T T' T (-)"n?" +~+ 2! 41 6! (2n)! A) B) 0 C) -12 D) E) T...
5 answers
P(A or B) assuming that P(A) $ 0.53,P(B) $ 0.43 and P(A and B) s 0.32.A.64 B.85 C.96 D.75
P(A or B) assuming that P(A) $ 0.53,P(B) $ 0.43 and P(A and B) s 0.32. A.64 B.85 C.96 D.75...

-- 0.019475--