$4.$ Design $-$calculus terms for the following functions: $($a$)(5$ points$)$ xy$[$ the maximum of x and y $]($b$)(5$ points$)$ xy$[$ the minimum of x and y $]($c$)(5$ points$)$ xyz$[$ if $($z $=0$ or z $=1)$ then $0$ else x $+$ y $($mod z$)]($d$)(4$ points$)$ x$[$ the number of $1$s in the binary representation of x $]($e$)(4$ points$)$ The characteristic function A of A where the set A is recursively defined as follows: $1.0$ in A $2.$ if n in A then $23$ n in A $3.$ A is the smallest set satisfying the above conditions.