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linuxux
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Hello, I'm working on some questions and I need some further explanation;
First I must Consider the open interval (0,1), and let S be the set of point in the open unit square; that's is, S={(x,y):0<x,y<1}.
Question (a) says Find a 1-1 function that maps (0,1) into, but not necessarily onto, S. (this is easy)
So my first thought is the function [tex](\frac{1}{a},\frac{1}{b}) : a,b\in\mathbf{N}[/tex]. I think this function is into, but not all the members of the set S are mapped to, thus not being onto. Is this right?
The next question i had a problem with: (b) Use the fact that every real number has a decimal expansion to produce a 1-1 function that maps S into (0,1). Discuss whether the formulated function is onto. (keep in mind that any terminating decimal expansion such as .235 represents the same real number .2349999... .)
Heres my thought for (b), i define a set [tex]A_n , n\in\mathbf{N}[/tex] where n represents the number of decimal places. So n=1 would be one decimal place, n=2 would be two decimal places, and so on. With only one decimal place, [tex]A_1[/tex] contains every combination of (x,y) such that 0<x,y<1 for instance (.9,.3) or (.1,.5). [tex]A_2[/tex] contains every combination of (x,y) such that 0<x,y<1 to two decimal places, and so on.
So if i arrange all the [tex]A_n[/tex] like this:
[tex]A_1(1)[/tex] [tex]A_1(2)[/tex] [tex]A_1(3)[/tex] [tex]A_1(4)[/tex] ...
[tex]A_2(1)[/tex] [tex]A_2(2)[/tex] [tex]A_2(3)[/tex] ...
[tex]A_3(1)[/tex] [tex]A_3(2)[/tex] ...
[tex]A_4(1)[/tex] ...
...
and i arrange [tex]\mathbf{N}[/tex] like this:
1 3 6 10 ...
2 5 9 ...
4 8 ...
7 ...
...
Then S maps into [tex]\mathbf{N}[/tex] since i represented every decimal expansion in the array. Is this right? But I'm not sure how the reminating decimal affects this, since, for instance, .9 which belongs to [tex]A_1[/tex] is also .8999... .
The puzzling thing about this question is that at the end of the question it says this: The Schroder-Bernstein Theorem discussed in 1.4.13 to follow can now be applied to conclude that (0,1)~S.
Why would i apply something that follows to this question?
First I must Consider the open interval (0,1), and let S be the set of point in the open unit square; that's is, S={(x,y):0<x,y<1}.
Question (a) says Find a 1-1 function that maps (0,1) into, but not necessarily onto, S. (this is easy)
So my first thought is the function [tex](\frac{1}{a},\frac{1}{b}) : a,b\in\mathbf{N}[/tex]. I think this function is into, but not all the members of the set S are mapped to, thus not being onto. Is this right?
The next question i had a problem with: (b) Use the fact that every real number has a decimal expansion to produce a 1-1 function that maps S into (0,1). Discuss whether the formulated function is onto. (keep in mind that any terminating decimal expansion such as .235 represents the same real number .2349999... .)
Heres my thought for (b), i define a set [tex]A_n , n\in\mathbf{N}[/tex] where n represents the number of decimal places. So n=1 would be one decimal place, n=2 would be two decimal places, and so on. With only one decimal place, [tex]A_1[/tex] contains every combination of (x,y) such that 0<x,y<1 for instance (.9,.3) or (.1,.5). [tex]A_2[/tex] contains every combination of (x,y) such that 0<x,y<1 to two decimal places, and so on.
So if i arrange all the [tex]A_n[/tex] like this:
[tex]A_1(1)[/tex] [tex]A_1(2)[/tex] [tex]A_1(3)[/tex] [tex]A_1(4)[/tex] ...
[tex]A_2(1)[/tex] [tex]A_2(2)[/tex] [tex]A_2(3)[/tex] ...
[tex]A_3(1)[/tex] [tex]A_3(2)[/tex] ...
[tex]A_4(1)[/tex] ...
...
and i arrange [tex]\mathbf{N}[/tex] like this:
1 3 6 10 ...
2 5 9 ...
4 8 ...
7 ...
...
Then S maps into [tex]\mathbf{N}[/tex] since i represented every decimal expansion in the array. Is this right? But I'm not sure how the reminating decimal affects this, since, for instance, .9 which belongs to [tex]A_1[/tex] is also .8999... .
The puzzling thing about this question is that at the end of the question it says this: The Schroder-Bernstein Theorem discussed in 1.4.13 to follow can now be applied to conclude that (0,1)~S.
Why would i apply something that follows to this question?
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