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Hello friends, could you help me with examples of bijective functions in pairs of intervals
a) [0,1) and (0,1)
b) (1,2) and (2,3)
c) [0,1) and (0,1]
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Bijective functions
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Re: Bijective functions
by HallsofIvy » Fri Jan 15, 2021 10:42 am
I wrote out a whole thing about what "bijective functions" are and then realized that wasn't your problem!
Your problem, and the reason why linear functions won't work is that, for the first and third at any rate, one interval is open and the other isn't. You can't just map end points to end points with a linear function. In fact, you can't use continuous functions!
For the first, I recommend mapping 0 to some point in the middle of (0, 1), say 1/2, then mapping points in (0, 1/2) to (0, 1/2) and points in [1/2, 1) to [1/2, 1].
The second exercise doesn't have that problem. f(x)= x+ 1 works nicely.
Your problem, and the reason why linear functions won't work is that, for the first and third at any rate, one interval is open and the other isn't. You can't just map end points to end points with a linear function. In fact, you can't use continuous functions!
For the first, I recommend mapping 0 to some point in the middle of (0, 1), say 1/2, then mapping points in (0, 1/2) to (0, 1/2) and points in [1/2, 1) to [1/2, 1].
The second exercise doesn't have that problem. f(x)= x+ 1 works nicely.
- HallsofIvy
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