Show that the following set has the same cardinality as $\mathbb R$ using CSB We have to show that the following set has the same cardinality as $\mathbb R$ using CSB (Cantor–Bernstein–Schroeder theorem).
$\{(x,y)\in \Bbb{R^2}\mid x^2+y^2=1 \}$
I think that these are the two functions:
$f:(x,y)\to \Bbb{R} \\f(x)=x,\\f(y)=y $
$g:\Bbb{R}\to (x,y)\\ g(x)=\cos(x),\\g(y)=\sin(y)$
Is this correct ? 
Thanks.

HINT: There is no continuous bijection between the two sets. Find a bijection from the unit circle to $[0,2\pi)$, and an injection from $\Bbb R$ into $[0,2\pi)$.
Also, when you define a function $g\colon\Bbb R\to\Bbb R^2$ you don't write $g(x)=\cos x$ and $g(y)=\sin y$. You should write $g(x)=(\cos x,\sin x)$ instead. Similarly when defining $f\colon\Bbb R^2\to\Bbb R$, you should define $f(x,y)=z$ rather than writing $f(x)=x$ and $f(y)=y$.
Both functions that you have defined are meaningless expressions.
