3

(7 marks) Suppose X countable_ and A 5 an infinite substt o X_ Show tbat councable, clearly stating any results JOu use from the course...

Question

(7 marks) Suppose X countable_ and A 5 an infinite substt o X_ Show tbat councable, clearly stating any results JOu use from the course

(7 marks) Suppose X countable_ and A 5 an infinite substt o X_ Show tbat councable, clearly stating any results JOu use from the course



Answers

Suppose that $A$ is a countable set. Show that the set $B$ is also countable if there is an onto function $f$ from $A$ to $B$ .

So we're given comfortably. Many sets a n which accountable, and we want to show that their union is still countable, so the countable union of comfortable sets is still accountable. So to do that, let's fry the diagram where we have in the first column the elements off the set a one. So let's call them a one. A two, a three and so forth, then another column. You write the elements off the set a two. Let's call them B one B two B three and so forth. Then another call on the sets of a three. C wants to see three and we completely the ground. The diagram In this way. Now here, let's assume that all the sets a narrow, infinitely comfortable and all the elements are different because that's the hardest case. Because if some of the sets a we're fine, it, then that's not a big deal. The union off finance sets you still finance, so that doesn't change anything. And if there are some other lap ings between the sets, so if one element is both in some a one and a two, well, when we're uniting them, the union is Moeller, then it will be if all the sets were different. So the set where all the kids were all the sets are infinitely comfortable And all these joint is the hardest possible case. So let's assume that So all these a wanted to be three and all these numbers are all different. So the union of all the sets a wanted to a three and so forth We decide that contains all these numbers They wanted to be one and so forth. So now we want a way to enumerates these numbers. But again here the trick is going by diagonal enumeration. So this first element here we call it X one. Then we proceed diagonally So these x two and we go diagonally This is X three and then we proceed by the diagonals disease called x four We going diagonals disease x five Keep going Diagonals, thes xx and so forth. So, by drawing these diagonals, we're leasing all the elements off this union. Now the union ofall the end is going to be the set that contains x one x two x tree and so forth. And because a while we've laid out this diagram all these ex Sonics. Two extra A Are all the elements off these union when out is an enumeration off the elements of the union, and so it is comfortable.

Let's take a final length strings off bit so that for instance, 10110 and we see these is a boundary number to do so. It's essentially like when we're coming in with decimal numbers, but instead of living units multiples of 10 multiple of hundreds. Here we have powers off, too, so the right most digit corresponds to While the units saw 2 to 0, the second digit corresponds to well, the first power off to so to the first and so on. And so this number 10110 It's a binary number means we have one time. The fourth power of two plus zero times the third power off two plus one time The second power of two plus one time the first power of two plus zero times the zero power of two. Now this is zero. He's well exactly two days before, there is nothing and these are 16. So all you know, the number 10110 as a boundary number is our decimal number 22 and my first son. If you take 22 you take the biggest power to that. You can find that is below the number. So inside 22 we have 16 which is the part of two. And that's the greatest one because the next power to with 32 said That's to be so. Take 16 you remove it and you keep doing that until you the composer number in power's off suit. So in a sense, every positive number can be written is in a unique way as a binary number. So if s the notes set off boundary strings off final find it lands. That's essentially all the positive integers written in mine ary. And so we have him up from s into the post the vintage er's, which is by objective because every positive in district in Britain, uniquely as a boundary number and before the cardinal ity of us is equal to the carnality of the positive integers, which means as as is comfortable


Similar Solved Questions

5 answers
Not Given include the rational Write 3 inequality final il 8 interval lain the Vincludes but does
not Given include the rational Write 3 inequality final il 8 interval lain the V includes but does...
5 answers
Joe Chemist discovers that a certain reaction has rate of 00 x 102 moleculeslsec at 278C and the activation energy is z0OkJlmole_ At what temperature should run the reaction obtain exactly Avogadro'5 Number; 6.02 * 10* moleculeslsec? (R =8.31 x 10 * kJlmole K)307+
Joe Chemist discovers that a certain reaction has rate of 00 x 102 moleculeslsec at 278C and the activation energy is z0OkJlmole_ At what temperature should run the reaction obtain exactly Avogadro'5 Number; 6.02 * 10* moleculeslsec? (R =8.31 x 10 * kJlmole K) 307+...
5 answers
Your Question 1 inconrectthe UDHmt5/96261aqaimthe function f(r;y) 6r yrerion 1
Your Question 1 inconrect the UDHmt 5/9626 1 aqaim the function f(r;y) 6r y rerion 1...
5 answers
Find an equabon 0r the line tangent to tne given 6x + 4x &3* =Drapningtangent Iin e on the sare set 0i x2}The equation for the tangent Iine "[email protected]
Find an equabon 0r the line tangent to tne given 6x + 4x & 3* = Drapning tangent Iin e on the sare set 0i x2} The equation for the tangent Iine "[email protected]
5 answers
20.00 mL of a HzSO4 solution with an unknown concentration a 0.1285 M NaOH solution . was titrated to a phenolphthalein endpoint with 47.05 mL of What is the concentration of the HzSO4 solution?[HzSO4]
20.00 mL of a HzSO4 solution with an unknown concentration a 0.1285 M NaOH solution . was titrated to a phenolphthalein endpoint with 47.05 mL of What is the concentration of the HzSO4 solution? [HzSO4]...
5 answers
Suppose that A is 2X2 matrix with eigenvectors 2 V1 = Vz=[1 2 which correspond to -3 4, = 3,42 = 2 respectively: Find the general solution to the system X' = AXnone2t 2 y=C1e= +C2 € e 3t [2] -33t 2 +C2 € e 2t y =C1e` 27 -33t 2 [1 2 y =C1e` +C2 E e 2t { +t -3 2 -33t 2 2t 1 y =C1e` +c2te -3 2
Suppose that A is 2X2 matrix with eigenvectors 2 V1 = Vz=[1 2 which correspond to -3 4, = 3,42 = 2 respectively: Find the general solution to the system X' = AX none 2t 2 y=C1e= +C2 € e 3t [2] -3 3t 2 +C2 € e 2t y =C1e` 27 -3 3t 2 [1 2 y =C1e` +C2 E e 2t { +t -3 2 -3 3t 2 2t 1 y =C1...
5 answers
Find the local maximum and minimum values and saddle point(s) of the function; If vou have three-dimensional graphing software graph the function with domain and viewpoint that reveal all the important aspects of the function_ (Enter your answers as comma-separated Iist: If an answer does not exist, enter DNE:) f(x, Y) exy(1 X-Y)local maximum value(s)Iocal minimum value(s)saddle point(s)(x,Y, 0) =
Find the local maximum and minimum values and saddle point(s) of the function; If vou have three-dimensional graphing software graph the function with domain and viewpoint that reveal all the important aspects of the function_ (Enter your answers as comma-separated Iist: If an answer does not exist,...
1 answers
Find the area under the given curve over the indicated interval. $$ y=\frac{2}{x} ; \quad[1,4] $$
Find the area under the given curve over the indicated interval. $$ y=\frac{2}{x} ; \quad[1,4] $$...
5 answers
(Type His olnt Winana paid 5155 for 4 1 cost 1 1 1 4unchon 1 1 Millad Kuw" 1 1 ~statt$ whcn sho I Younq ad Msits Ihe z00 # HaueIAA 1 1 9 59!$ njjesllcd( 1 [ H ui 1
(Type His olnt Winana paid 5155 for 4 1 cost 1 1 1 4unchon 1 1 Millad Kuw" 1 1 ~statt$ whcn sho I Younq ad Msits Ihe z00 # HaueIAA 1 1 9 59!$ njjesllcd( 1 [ H ui 1...
5 answers
For Ihe following exercises graph Ihe nquations and shade the area 0l tha reglon between Ihe curves. Daterrnine Its araa by inteqrating over [he *MIs 20 ^ = j and. = 3y-2
For Ihe following exercises graph Ihe nquations and shade the area 0l tha reglon between Ihe curves. Daterrnine Its araa by inteqrating over [he *MIs 20 ^ = j and. = 3y-2...
1 answers
A compound microscope has the objective and eyepiece mounted in a tube that is 18.0 $\mathrm{cm}$ long. The focal length of the eyepiece is $2.62 \mathrm{cm},$ and the near-point distance of the person using the microscope is 25.0 $\mathrm{cm} .$ If the person can view the image produced by the microscope with a completely relaxed eye, and the magnification is $-4525,$ what is the focal length of the objective?
A compound microscope has the objective and eyepiece mounted in a tube that is 18.0 $\mathrm{cm}$ long. The focal length of the eyepiece is $2.62 \mathrm{cm},$ and the near-point distance of the person using the microscope is 25.0 $\mathrm{cm} .$ If the person can view the image produced by the mi...
4 answers
Solve for u(u+7)2 = 2u2 + 22u+61If there is more than one solution, separate
Solve for u (u+7)2 = 2u2 + 22u+61 If there is more than one solution, separate...
5 answers
Let f(x) is;and 9 (x) =Then (f.9) (-4)1 The value 32* 18 ;che exponential equation:U9 ?Express the logarithmic function logs Zy=r exponential for 20: - A " - 2 .5* {721 . loz {= 5 1og 10g 1oglog og log YPg: 4
Let f(x) is; and 9 (x) = Then (f.9) (-4) 1 The value 32* 18 ; che exponential equation: U9 ? Express the logarithmic function logs Zy=r exponential for 20: - A " - 2 .5* {7 21 . loz {= 5 1og 10g 1og log og log Y Pg: 4...
5 answers
Match the cultural or technological change with the species thatdeveloped it FIRST. Not all species names may be used, and any species namemay be used more than once BUT there is only ONE correct answer foreach question.41. burial of dead: A B C D E a) H. sapiens42. stone tools: A B C D E b) N
Match the cultural or technological change with the species that developed it FIRST. Not all species names may be used, and any species name may be used more than once BUT there is only ONE correct answer for each question. 41. burial of dead: A B C D E ...
5 answers
Researchers reported the results of a study in which 150 8year-old children completed a reading comprehension test (scoredout of 100 points). On this test, a higher score indicates greaterreading comprehension. On average, the children scored a 75.Identify for this study the (a) population, (b) sample, (c)dependent variable, and (d) descriptive statistic.
Researchers reported the results of a study in which 150 8 year-old children completed a reading comprehension test (scored out of 100 points). On this test, a higher score indicates greater reading comprehension. On average, the children scored a 75. Identify for this study the (a) population, (b) ...
5 answers
For a hypothesis testing problem with a given data set, you have rejected the null hypothesis at a 10% level of significance. For the same data set,
For a hypothesis testing problem with a given data set, you have rejected the null hypothesis at a 10% level of significance. For the same data set,...

-- 0.020705--