I need to prove the following:
a) Prove that every closed interval [a,b] is numerically equivalent to [0,1].
b) Prove that any two closed intervals [a,b] and [c,d] are numerically equivalent.
c) Prove that any two open intervals (a,b) and (c,d) are numerically equivalent.
d) Let [a,b] be a closed interval and (c,d) be an open interval. Prove that [a,b] and (c,d) are numerically equivalent.
I'm guessing all of these are proven pretty much the same way, but I don't even know where to start. Can anyone help?
Can you define Numerically Equivalent?
Let A and B be sets contained in some universal set U. We say that A and B are numerically equivalent if there exists a bijection f: A->B.
It should be obvious that the easiest map to make has a going to 0 and b going to 1. So f(a) = 0 and f(b) = 1.
Using this logic, we can then say that mid points map to mid points. So f((a-b)/2) = 1/2. Continue this logic, and you will eventually get every point in [a, b].
So doing some guess and checking, I come up with: f(c) = (c - a)/(b - a)
Note that a and b are constants, thus, this is just a linear map. Your teacher may or may not allow you to use the fact that all linear maps are bijective. Also note that it fails for when a = b, as expected.
b follows the same exact logic as a. But, instead of having from 0 to 1, you have from c to d.
I'll take a look at c and d later.