For the following circuits put them into Product of Maxterms

For the following circuits, put them into Product of Maxterms and Sum of Minterms forms. I do not care which of the two ways I mentioned you use to do it (make a truth table, or manipulate into product of sums/sum of products form). Remember that for complex equations, you may want to evaluate subfunctions first. G = A^OverBar * B * C^OverBar * D + D *C^OverBar (A^OverBar + B^OverBar) + A^OverBar * C(B^OverBar + D^OverBar) + A * B^OverBar * C^OverBar * D F =((A CirclePlus D) * C^OverBar + B) * (A^OverBar *B + C * D^OverBar^OverBar)

Solution

a) Given function is

G = A\'BC\'D + DC\'(A\'+B\') + A\'C(B\'+D\') + AB\'C\'D

Application of Distributive property

A\'BC\'D + DC\'A\' + DC\'B\' + A\'CB\' + A\'CD\' + AB\'C\'D

Creating Min terms

A\'BC\'D + DC\'A\'(B + B\') + DC\'B\'(A + A\') + A\'CB\'(D + D\') + A\'CD\'(B + B\') + AB\'C\'D

Removing redundant terms

A\'BC\'D + A\'BC\'D + A\'B\'C\'D + AB\'C\'D + A\'B\'C\'D + A\'B\'CD + A\'B\'CD\' + A\'BCD\' + A\'B\'CD\' + AB\'C\'D

Sum of Min Terms

G(A,B,C,D) = A\'BC\'D +  A\'B\'C\'D + AB\'C\'D + A\'B\'CD + A\'B\'CD\' + A\'BCD\'

Above equation is the sum of min terms equation.

Product of max terms could be easily found by taking missing min terms in G and apply complement on that function using demorgan\'s theorem.

Let the missing min terms are written as the function

X(A,B,C,D) = A\'B\'C\'D\' + A\'BC\'D\' + A\'BCD + AB\'C\'D\' + AB\'CD\' + AB\'CD + ABC\'D\' + ABC\'D + ABCD\' + ABCD

Product of max terms is obtained by complementing the function X.

X\'(A,B,C,D) = A\'B\'C\'D\' * A\'BC\'D\' * A\'BCD * AB\'C\'D\' * AB\'CD\' + AB\'CD + ABC\'D\' + ABC\'D + ABCD\' + ABCD

Product of Max Terms

X\'(A,B,C,D) = (A+B+C+D) * (A+B\'+C+D) * (A+B\'+C\'+D\') * (A\'+B+C+D) * (A\'+B+C\'+D) * (A\'+B+C\'+D\') * (A\'+B\'+C+D) * (A\'+B\'+C+D\') * (A\'+B\'+C\'+D) * (A\'+B\'+C\'+D\')