## Injection from R2 to R

For college and university level advanced Mathematics
nayel
Posts: 268
Joined: Tue Dec 07, 2010 7:38 pm
Location: Dhaka, Bangladesh or Cambridge, UK

### Injection from R2 to R

Find an injection from $\mathbb R^2\to\mathbb R$.
"Everything should be made as simple as possible, but not simpler." - Albert Einstein

tanvirab
Posts: 446
Joined: Tue Dec 07, 2010 2:08 am

### Re: Injection from R2 to R

Interesting. A classic space-filling curve should work. Are there easier solutions?

nayel
Posts: 268
Joined: Tue Dec 07, 2010 7:38 pm
Location: Dhaka, Bangladesh or Cambridge, UK

### Re: Injection from R2 to R

Should be, as it was on one of our examples sheets. And I don't even know what that term means
"Everything should be made as simple as possible, but not simpler." - Albert Einstein

Avik Roy
Posts: 156
Joined: Tue Dec 07, 2010 2:07 am

### Re: Injection from R2 to R

There's one that maps all the points in $(0,1) \times (0,1)$ to $(0,1)$. Take $(0.a_1a_2a_3 ... ,0.b_1b_2b_3 ...) \rightarrow 0.a_1b_1a_2b_2a_3b_3...$. Such a map should work also
"Je le vois, mais je ne le crois pas!" - Georg Ferdinand Ludwig Philipp Cantor

nayel
Posts: 268
Joined: Tue Dec 07, 2010 7:38 pm
Location: Dhaka, Bangladesh or Cambridge, UK

### Re: Injection from R2 to R

Yep that should work.
"Everything should be made as simple as possible, but not simpler." - Albert Einstein