Example $2.$I$\mathrm{.}9$ restates Aristotle$$s Paradox as: the intervals I$0=[0,2\backslash$pi r$0)$ and I$1=[0,2\backslash$pi r$1)$

have the same cardinality for r$0,$ r$1$ in R$+.$

$($a$)$ Verify it by checking that g : I$0->$ I$1$ given by g$($x$)=$ x $($r$1/$r$0)$ is a correspondence.

$($b$)$ Show that when a $<$ b$,$ the cardinality of $[0,1)$ equals that of $[$a$,$ b$).$

$($c$)$ Generalize by showing that when a $<$ b and c $<$ d$,$ that the intervals $[$a$,$ b$)$ and $[$c$,$ d$)$

have the same cardinality.