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

Answer 1

If you want to map (a,b) -> c, such that no two different (a,b) map to the same c, why not just interlace the digits. So (375,98) = (375,098) -> 307958, and (1/3,1/5) = (.333..., .2) -> .323030.., and so on. I believe this is a valid injection.

Answer 2

Some care is needed in Ben's answer to deal with numbers that have more than one decimal expansion, but the idea does work. The function obtained is not onto, however. Also, it is fun to find an injection from R to R^2.

Answer 3

Are you looking for an example?

"In mathematics, an injective function is a function which associates distinct arguments to distinct values. More precisely, a function f is said to be injective if it maps distinct x in the domain to distinct y in the codomain, such that f(x) = y."

So an "injection" is just another name for a one-to-one (every output can be produced by only one input) function, and mapping from R² to R just means that the function has to take two real numbers as an input, and has to produce one real number as an output.

Any one-to-one function of two variables that you can write as f(x, y) would qualify.

I guess, after thinking about it, it's a little tougher to make a function of 2 variables one-to-one....trying to think of one.

Ugh...this is harder than I thought...

Yeah, well, I guess Ben's works. :c( I tried.

Answer 4

What do you mean by injection? I think it would be helpful if you could edit your question and clarify a bit - thanks.

Answer 5

Ben is right.