$($let$\text{'}$s practice functional decomposition more$)$ Consider the following function:

$f(a,b,c,d,e)={\displaystyle \underset{?}{\overset{?}{}}}m(0,2,5,7,9,11,12,14,16,18,21,23,25,27,28,30)$

a$)$ Use a Karnaugh map to produce the optimal SOP representation of this function. Create the

list of all AND, OR$,$ and NOT gates you would need to implement this circuit, including the

number of inputs for each AND and OR gate. Calculate the cost of implementation using the cost

function as defined in the textbook and HW $3.$

b$)$ Use functional decomposition $($pattern recognition in a Karnaugh Map$)$ to find the simplest

implementation of this function $($give us a Boolean equation$),$ and show how it would be

implemented using $2-$input lookup tables $($LUT$).$ Draw a circuit consisting of only $2-$input LUTs,

and make sure to show the content of each LUT and label all inputs and outputs. Only non$-$

inverted inputs a$,$ b$,$ c$,$ d$,$ and e are available. Note: you should be able to easily see how your

result from a$)$ could be modified to match what you have produced in b$).$