Showing two sets are equivalent

[hwill205]

Homework Statement

Show that R [tex]\approx[/tex] R⁺ , that is, the set of all real numbers is equivalent to the set of all positive real numbers

Homework Equations

The only relevant equation is finding one such that F:R[tex]\rightarrow[/tex]R⁺ is a bijection.

The Attempt at a Solution

I've attempted to use the tangent function as a bijection, but that doesn't work. This isn't for homework but for a test review sheet, so all guidance is gladly appreciated.

 

[Wingeer]

Equivalent? You could probably show that the sets are isomorphic by finding an isomorphism. The function:
[tex]F: \mathbb{R} \to \mathbb{R}^+[/tex]
[tex]F:x \to x^2[/tex]
would map the real numbers into positive real numbers. I should think F is an isomorphism over these sets.

 

[hwill205]

But that isn't a bijection as its not one-to-one. For example, -5 maps to 25 and 5 maps to 25. If I can find a bijection from R to R+ ,then I've proven they are equivalent sets. Just can't find the darn bijection.

 

[micromass]

Hi hwill205

You should think of exponential functions...

 

[hwill205]

Damn, y=e^x works. Thanks a lot man. Can't believe I didn't think of that.

 

[Wingeer]

But that isn't a bijection as its not one-to-one. For example, -5 maps to 25 and 5 maps to 25. If I can find a bijection from R to R+ ,then I've proven they are equivalent sets. Just can't find the darn bijection.

Oh, of course. That + sign made me think of positive real numbers, and I messed up codomain and domain. :(