F (X,Y,Z) = ΠM )7,6,5,3( Explicit Compact max-term form for 0s of the function
F(X,Y,Z) = Π )7,6,5,3(^ Implicit Compact max-term form for 0s of the function
Relationship between Min-terms and Max-terms
Min-terms and Max-terms are complements of each other : Mi=mi and Mi=mi
DeMorgan’s Theorem is key to proving the min-term/max-term relationship:a) X 1 +X 2 +K+Xn=X 1 • X 2 • K•Xn
b) X 1 • X 2 • K•Xn=X 1 +.X 2 .+K+Xn
Examples:
Given max-term M 6 = X+.Y+Z, find min-term m 6.1) Since it is a max-term, when X=1, Y=1 and Z=0 Then F(X,Y,Z) = 0
2) To convert to Min-term we can apply DeMorgan’s Theorem which in practice is dividing up
the overbar. This means that the cross bar can be divided across its subpart while accepting the
rules:+=. and .=+
Let’s see how it applies to our example.We know that Mi=mi and Mi=mi so
Min −term =mi=X+.Y+Z=X+Y+Z=XY.. Z
Example: Apply the overbar to finding Complement of F if F(X,Y,Z) = (X+.Y).(Y+Z)
Solution:
Apply the DeMorgan’s Theorem in the form of “Dividing up the Overbar”.
F(X Y,, Z)=(X+Y).(Y+Z)=(X+Y (.) Y+Z)=(XY). +YZ).( =X.Y+Y.Z Example
Write Standard SOP and POS form for f(x 3 , x 2 , x 1 , x 0 ) = ∑ )15,12,7,0(
Solution:Converting between compact forms of functions