set theory problem, open and close intervals equivalance

Nov 2013
6
0
The City of Eagles
hello there,
i am trying to prove a set theory problem relating open and half open intervals. help needed.
1- [0,1] ~ (a,b)
2- [0,1] ~ [a,b)
3- [0,1] ~ (a,b]
where a,b belongs to R and ~ is equivalent sign.

for question 2, [0,1] ~ [a,b)
i have defined a function such that f(x) = (bx+a/2) / 2 for all x belongs to [0,1]. is that correct??
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Two sets are "equivalent" if and only if there exist an invertible function that maps every point in the first set to every point in the other set. Your first problem requires that you find an invertible function that maps (0, 1) to (a, b). You want to use f(x)= (bx+ a/2)/2 which is invertible. But f(0)= a/4 and f(1)= b/2+ a/4. That does NOT map (0, 1) to (a, b).

But there is a more serious problem. To do this problem you should know that a continuous invertible function must map open sets to open sets and closed sets to closed sets. Since [0, 1] is closed but none of (a, b), [a, b), or (a, b] are closed, no continuous, and certainly no linear function will work. I would recommend something like: map every irrational number to itself and then use the fact that the set of rational numbers in both [0, 1], (a, b), [a, b), (a, b] are countable.
 
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Nov 2013
6
0
The City of Eagles
Two sets are "equivalent" if and only if there exist an invertible function that maps every point in the first set to every point in the other set. Your first problem requires that you find an invertible function that maps (0, 1) to (a, b). You want to use f(x)= (bx+ a/2)/2 which is invertible. But f(0)= a/4 and f(1)= b/2+ a/4. That does NOT map (0, 1) to (a, b).

But there is a more serious problem. To do this problem you should know that a continuous invertible function must map open sets to open sets and closed sets to closed sets. Since [0, 1] is closed but none of (a, b), [a, b), or (a, b] are closed, no continuous, and certainly no linear function will work. I would recommend something like: map every irrational number to itself and then use the fact that the set of rational numbers in both [0, 1], (a, b), [a, b), (a, b] are countable.
would you mind giving any examples of f(x) which can be used in these problem. i am new to this stuff, dnt have much grip. i'll much obliged.
 

emakarov

MHF Hall of Honor
Oct 2009
5,577
2,017
would you mind giving any examples of f(x) which can be used in these problem.
Have you seen the tread I linked to? Do you think I kept in my bookmarks for 1.5 years in vain? :)
 
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Plato

MHF Helper
Aug 2006
22,507
8,664
i am trying to prove a set theory problem relating open and half open intervals. help needed.
1- [0,1] ~ (a,b)
2- [0,1] ~ [a,b)
3- [0,1] ~ (a,b]
I see that you have solved this question. Here is some advice for future questions like this.
You can show that \(\displaystyle [0,1]\sim (0,1]\sim [0,1)\sim (0,1)\).
We do that by point shifting of a countable set. Define a sequence \(\displaystyle f_0=0,~f_n=n^{-1},~n\in\mathbb{Z}^+\).

Now if \(\displaystyle x\in [0,1]\) and if \(\displaystyle (\exists k)[x=f_k]\) then define \(\displaystyle \pi:~x\mapsto f_{k+1}\) or else \(\displaystyle \pi:~x\mapsto x\).
That shows that \(\displaystyle [0,1]\overset{\pi}\sim(0.1]\). Similar functions made with minor adjustments in \(\displaystyle \pi\).

Note that if \(\displaystyle a<b\) then \(\displaystyle \sigma: x\mapsto a+(b-a)x\) makes \(\displaystyle [0,1]\overset{\sigma}\sim [a,b]\).
 
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