Let a and b be real numbers. Consider the statement $$If a $>0$ and b $<0,$ then ab $<0.$

$($a$)$ Rewrite the statement as a proposition using predicate variables p$,$ q$,$ and r$.$ Be sure to

define p$,$ q$,$ and r$.$

$($b$)$ Determine whether we can make the following conclusions, assuming the statement

above is true. Explain why or why not.

$($i$)$If ab $<0,$ then a $>0$ and b $<0.$

$($ii$)$Given that ab $>=0.$ Then, a $<=0$ or b $>=0.$