Question 13.16: UsingDon’t Cares to Simplify Expressions—1 Use don’t care en...
UsingDon’t Cares to Simplify Expressions—1
Use don’t care entries to simplify the expression:
f(a, b, c, d)=\bar{a} \cdot \bar{b} \cdot \bar{c} \cdot d+\bar{a} \cdot \bar{b} \cdot c \cdot \bar{d}+\bar{a} \cdot \bar{b} \cdot c \cdot d+\bar{a} \cdot b \cdot \bar{c} \cdot d+a \cdot \bar{b} \cdot c \cdot d+a \cdot b \cdot \bar{c} \cdot \bar{d}
Known Quantities: Logical expression; don’t care conditions.
Find: Minimal realization.
Schematics, Diagrams, Circuits, and Given Data: Don’t care conditions:
f(a, b, c, d)=\{0100,0110,1010,1110\}
Analysis: We cover the Karnaugh map of Figure 13.52 using ones, and also using x entries for each don’t care condition. Treating all of the x entries as ones, we complete the covering with two four-cell subcubes and one two-cell subcube, to obtain the following simplified expression:
f(a, b, c, d)=b \cdot \bar{d}+\bar{b} \cdot c+\bar{a} \cdot \bar{c} \cdot d
Comments: Note that we could have also interpreted the don’t care entries as zeros and tried to solve in product-of-sums form. Verify that the expression obtained above is indeed the minimal one.
