Final answer:
To simplify the Boolean functions using four variables K-maps, we represent the functions in K-maps and group the 1s to form the minimum sum of products (MSP) and minimum product of sums (MPOS) expressions. The MSP and MPOS expressions for the given functions are provided.
Step-by-step explanation:
To simplify the Boolean functions using four variables K-maps, we first need to represent the given functions in K-maps. Then, we will group the 1s in the K-map to form the minimum sum of products (MSP) and minimum product of sums (MPOS) expressions.
a. F(A,B,C,D) = ∑m(0,1,2,4,5) + d(3,6,7)
In the K-map, we can see that the 1s can be grouped as follows:
AB'C'D' + AB'CD' + ABC'D' + ABCD + AB'CD
So, the MSP expression is: F(A,B,C,D) = AB'C'D' + AB'CD' + ABC'D' + ABCD + AB'CD
The MPOS expression is: F(A,B,C,D) = (A+B+C+D)(A+B+C'+D')(A+B'+C+D')(A+B'+C+D)(A+B'+C+D')
b. F(X,Y,Z,W) = ΠM(0,6,8,13,14) + d(2,4,10)
In the K-map, we can see that the 0s can be grouped as follows:
X'YZ'W' + X'Y'ZW' + X'YZW' + XYZ'W'
So, the MSP expression is: F(X,Y,Z,W) = X'YZ'W' + X'Y'ZW' + X'YZW' + XYZ'W'
The MPOS expression is: F(X,Y,Z,W) = (X+Y'+Z)(X'+Y+Z)(X+Y+Z')(X+Y+Z)
c. F(A,B,C,D) = ∑m(4,6,7,8,12,15) + d(2,3,5,10,11,14)
In the K-map, we can see that the 1s can be grouped as follows:
AB'CD' + ABC'D' + ABCD' + ABCD + AB'CD + AB'CD'
So, the MSP expression is: F(A,B,C,D) = AB'CD' + ABC'D' + ABCD' + ABCD + AB'CD + AB'CD'
The MPOS expression is: F(A,B,C,D) = (A+B+C+D')(A+B'+C+D')(A+B'+C'+D')(A+B'+C+D)(A+B'+C+D)