$4.[12$ pts$]$ Consider a structured space which corresponds to selecting $\backslash ($ k $\backslash )$ or more items from a set of $\backslash ($ n $\backslash )$ items, where $\backslash ($ n $\backslash$geq $1\backslash )$ and $\backslash (0\backslash$leq k $\backslash$leq n $\backslash ).$ A Boolean formula $\backslash (\backslash$Delta $\backslash )$ captures this space iff there is a one$-$to$-$one correspondence between the possible selections and the satisfying assignments of $\backslash (\backslash$Delta $\backslash ).$ Suppose we use the Boolean variable $\backslash ($ A$\_\{$i$\}\backslash )$ to indicate whether item $\backslash ($ i $\backslash )$ is selected.

$(4$pt$)$ Describe a CNF that captures this structured space. How many clauses does the CNF have in terms of $\backslash ($ n $\backslash )$ and $\backslash ($ k $\backslash ).$

$(4$pt$)$ Describe a DNF that captures this structured space. How many terms does the DNF have in terms of $\backslash ($ n $\backslash )$ and $\backslash ($ k $\backslash ).$

$(4$pt$)$ Can you capture this structured space more efficiently using an OBDD? If so$,$ describe the OBDD and its size in terms of $\backslash ($ n $\backslash )$ and $\backslash ($ k $\backslash ).$