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incompletely specified function - Boolean Algebra
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An Incompletely specified function is a Boolean function that only define output values for a subset of its inputs - i.e. a Boolean function whos output is a don't care for at least one of its input combinations. Incompletely specified functions often make no guarantees as to the unspecified output whatsoever.


A B C Equation f
0 0 0 1
0 0 1 X
0 1 0 X
0 1 1 1
1 0 0 1
1 0 1 0
1 1 0 0
1 1 1 X

In many situations when working with combinational circuits, some combinations of inputs should not occur under normal working conditions. For circuits with such combinations, those combinations can be treated as either 0 or 1 depending on whichever yields a more simplified Boolean expression.

Boolean functions with don't care output terms are represented with a Equation d . For example consider the following function.

Equation f left-parenthesis upper A comma upper B comma upper C right-parenthesis equals sigma-summation m left-parenthesis 0 comma 3 comma 4 right-parenthesis plus d left-parenthesis 1 comma 2 comma 7 right-parenthesis

Which can be represented with the following truth table and K-map.

kmap unspecified func example.svg

Note that depending on the final design of the circuit, the output for Equation f left-parenthesis 0 comma 0 comma 1 right-parenthesis for example may be either 0 or 1.


kmap (don't care example).svg
Main article: karnaugh_map#Don't cares

Karnaugh maps are especially suitable for minimizing logic circuts with don't care outputs because they directly translate to don't care cells on a k-map. If grouping don't care cells results in larger groups, then more variables may be eliminated resulting in smaller Boolean expressions.


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