Do exist an injective function from $\mathbb{R}^5 \rightarrow \mathbb{R}^4$?
I think it is true, but I am unable to find a simple example... Please help me.
I think it should be something like $$ q:\mathbb{R}^5 \rightarrow \mathbb{R}^4, \ q(x,y,z,k,l) = (f(x,y),\ f(y,z),\ f(z,k),\ f(k,l)); $$ where $f:\mathbb{R}^2\rightarrow\mathbb{R}$, an injective function, as I just read here about that: Injective function from $\mathbb{R}^2$ to $\mathbb{R}$?
Am I wrong?
If there exists a such function, please give me an example. Thank you very much!
You can use the injection from $\Bbb R^2 \to \Bbb R$ that you read about. If that bijection is $m=f(x,y)$ you can take $(x,y,z,k,l) \to (m,z,k,l)$ The last three components just go along for the ride.