Incompletely Specified Functions

- Incompletely specified function:

- Output value is not known for every combination of input variables because all combinations cannot occur.

- Truth table does not generate an output value for every possible combination of input values.

- Min-terms or max-terms that are not used as part of the output function are called don’t care terms.

Examples

Conversion of BCD to Ex-3

Binary				Ex-3
W	X	Y	Z	A	B	C	D
0	0	0	0	0	0	1	1
0	0	0	1	0	1	0	0
0	0	1	0	0	1	0	1
…	…	…	…	…	…	…	…
1	0	0	1	1	1	0	0
1	0	1	0	Don’t care
1	0	1	1	Don’t care
…	…	…	…	…	…	…	…
1	1	1	1	Don’t care

A = f(w,x,y,z) = Σ(5,6,7,8,9) + Σ d(10,11,12,13,14,15)

B = f(w,x,y,z) = Σ(1,2,3,4,9) + Σ d(10,11,12,13,14,15)

C = f(w,x,y,z) = Σ(0,3,4,7,8) + Σ d(10,11,12,13,14,15)

D = f(w,x,y,z) = Σ(0,2,4,6,8) + Σ d(10,11,12,13,14,15)

- Don’t care terms can be assigned a value 0 or 1.

Rule for simplification with K-Map

If don’t care terms help in creating a larger group of min-terms, then we can assign it as a 1. If it does not help in creating a larger group, then we assign it as a 0.

Example

A = W + XZ + XY

B = X’Z + X’Y + XY’Z’

C = Y’Z’ + YZ

D = Z’

A = (W’ * (XZ)’ * (XY)’)’

B = ((X’Z)’ * (X’Y)’ * (XY’Z’)’)’

C = ((YZ)’ * (Y’Z’)’)’

D = Z’