1-1 correspondances in R

RolandIRL

how would you show that a bijection (or 1-1 correspondances) exists between subsets of R?
there's this question in a problem sheet (not for homework) that asks to set up a simple bijection to show that cardinality of (-pi/2,pi/2) equals the cardinality of R.
i know how to set up bijections for N,Z and Q, but the lecturer never really went into much detail on R. the lecturer wants us to figure it out on our own.
Thanks

Fremen

Funny, this exact point came up in a discussion I had yesterday.
You need a continuous, 1-1 function which flies off to infinity at pi/2, and flies off to minus infinity at -pi/2.
What well-known function does that?



There are probably other ways you could do it, too. for example, show that the cardinality of (-pi/2,pi/2) is equal to that of (-pi,pi), and then use an induction argument.

Fremen

Incedentally, you can show that the cardanality of R is strictly greater than the cardinality of Q. This raises the following question:

"does there exist a set S such that, with strict inequality
Card(Q) < Card(S) < Card(R)"?

The answer is goddamn freaky. Ask your lecturer about it.

RolandIRL

thanks fremen. i doubt i would have got that on my own.
hmmm, i wonder what the function that tends to infinity as x-> pi/2 is....
it's just hard to associate trig with cardinality. the lecturer didn't even give any hints as to the answer.
thanks again fremen :)

Fremen

Yeah, that would have been a tough one if you hadn't seen something like it before. Whenever you see pi/2, pi/3, pi/6 etc, you should usually look for a trig interpretation.