## Cardinality of real numbers and set of real numbers between 0 and 1

**Question : **Show that the set of all real numbers has the same cardinality as the set of real number between 0 and 1.

**Solution : **Solution Let S be the open interval of real numbers between 0 and 1

S={x ∈ R| 0 < x < 1 }.

Imagine picking up S and bending it into circle as shown below. Since S does not include either endpoint 0 or 1, the top-most point of the circle is omitted from the drawing.