Question

1. Use the K-map to determine the prime implicants, essential prime implicants, a minimum sum of products, prime implicates, essential prime implicates, and a minimum product of sums for each of the following Boolean functions. Also, for each one compute a minimum product of sums and a minimum sum of products of its complements.

a. f(a,b,c,d)= Π M(0,1,8,11,12,14)
b. g(a,b,c,d)= Σ m(0,1,3,5,6,8,11,13,15)
c. h(a,b,c)= Σ m(1,4,5,6)
2. Write the decimal representation of SSOP and SPOS for each of the above functions and its complement.

Answer 1

To determine prime implicants, essential prime implicants, minimum sum of products, prime implicates, essential prime implicates, and minimum product of sums for Boolean functions, a Karnaugh map (K-map) is often used. The decimal representation of SSOP and SPOS for each function and its complement can be found by converting the Boolean functions into canonical forms.

Step-by-step explanation:

Boolean Functions:

a. There are multiple ways to find the prime implicants, essential prime implicants, minimum sum of products, prime implicates, essential prime implicates, and minimum product of sums for a Boolean function. One common method is to use a Karnaugh map (K-map) which is a graphical representation of a truth table. For function f(a,b,c,d), the prime implicants are BC' and AC' and the essential prime implicant is AC'D'. The minimum sum of products is (BC' + AC'D'). Similarly, the prime implicates, essential prime implicates, and minimum product of sums for the complement of function f(a,b,c,d) can be found.

b. For function g(a,b,c,d), the prime implicants are AB'CD', A'BCD', and AB'CD. The essential prime implicant is AB'CD'. The minimum sum of products is (AB'CD' + A'BCD' + AB'CD). Similarly, the prime implicates, essential prime implicates, and minimum product of sums for the complement of function g(a,b,c,d) can be found.

c. For function h(a,b,c), the prime implicants are A'B' and AB. The essential prime implicant is A'B'. The minimum sum of products is (A'B' + AB). Similarly, the prime implicates, essential prime implicates, and minimum product of sums for the complement of function h(a,b,c) can be found.

Decimal Representation:

To find the decimal representation of SSOP (Sum of SOP) and SPOS (SOP of Sum) for each Boolean function and its complement, we need to convert the Boolean functions into canonical forms. The SSOP is a disjunction of minterms and the SPOS is a conjunction of maxterms. The decimal representation of SSOP and SPOS can be found using the truth tables or K-maps.