how to find an injection from R^2 to R ?

Question

there is an injection from N^2 to N
(a,b) -> (3^a)(5^b) but i can't extend it to R.

Answer 1

I think there's no such thing... but I'm quite willing to be prooved wrong. It's feeling like a long time since my degree days...

I can do R+^2 to R, and even RxR+ to R using a function that takes alternate digits from each of the inputs (eg f(123,456)=142536 ) but I can't see a way to extend that to R^2.

Was this a trick question? ;)

*Edit* With regards to the answer below, it is true that finite sets cannot inject into a (strict) subset (this is obvious as there are more elements in the set than the subset) this isn't neccessarily true for infinite sets. The questioner gives a good example in the further details.

I'm curious to see the answer to this - are you sure such an injection exists?

Answer 2

not possible. R is a subset of R^2. any function R^2:R can not be one to one (definition of injection)