I am trying to show that the cardinality of the space between $x^2+y^2<1$ and $x+y>1$ is the same as the cardinality of real numbers
I haven't a clue where do begin with something like this. I have never worked with a space like this. Could someone lead me on the right direction please?
Your shape is a circle with the diamond in the center removed. Clearly, there exists a short line segment lying in this shape. There is an injective function from the real line into this little line segment. She shape is shown in blue here, and the little line segment is in red.
Let $A$ be the blue shape, and let the function $h:A\to\mathbb R^2$ be the inclusion function, $h(x)=x$. Since $\mathbb R^2$ has the same cardinality as $\mathbb R$, there is a bijection $f:\mathbb R^2\to\mathbb R$. Thus, $f\circ h$ is an injective function from $A$ to $\mathbb R$.
Since we have injective functions from $A$ to $\mathbb R$ and $\mathbb R$ to $A$, we can apply the Schroder-Bernstein (or Cantor-Bernstein) theorem to see that $\mathbb R$ and $A$ have the same cardinality.