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CD 00 00 01 11 10 0 0 0 0 01 0 0 1 0 11 0 0 0 1 10 0 0 0 0
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Don t Care Conditions Another simpli cation technique may be employed whenever the value of the logic function to be implemented can be either a 1 or a 0 This condition may result from the speci cation of the problem and is not uncommon Whenever it does not matter whether a position in the map is lled by a 1 or a 0, we use a so-called don t care entry, denoted by an x Then the don t care can be used as either a 1 or a 0, depending on which results in a greater simpli cation (ie, helps in forming the smallest number of maximal subcubes) The following examples illustrate the use of don t cares
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EXAMPLE 1316 Using Don t Cares to Simplify Expressions 1
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Use don t care entries to simplify the expression: 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 + a b c d
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Digital Logic Circuits
Note that the x s never occur, and so they may be assigned a 1 or a 0, whichever will best simplify the expression cd ab 00 01 11 10 00 0 x 1 0 acd bd 01 1 1 0 0 11 1 0 0 1 bc 10 1 x x x
Solution
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 1352 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 simpli ed expression: f (a, b, c, d) = b d + b c + a c 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
EXAMPLE 1317 Using Don t Cares to Simplify Expressions 2
Problem
Find a minimum product-of-sums realization for the expression f (a, b, c)
Solution
Known Quantities: Logical expression, don t care conditions Find: Minimal realization Schematics, Diagrams, Circuits, and Given Data:
f (a, b, c) = 1 for {a, b, c} = {000, 010, 011} f (a, b, c) = don t care for {a, b, c} = {100, 101, 110}
bc a 0 1 00 1 x 01 0 x 11 1 0 10 1 x
Analysis: We cover the Karnaugh map of Figure 1353 using ones, and also using x
entries for each don t care condition By appropriately selecting two of the three don t-care entries to be equal to 1, we complete the covering with one four-cell subcube and one two-cell subcube, to obtain the following minimal expression: f (a, b, c) = a b + c
Comments: Note that we have chosen to set one of the don t care entries equal to zero, since it would not lead to any further simpli cation
EXAMPLE 1318 Using Don t Cares to Simplify Expressions 3
Problem
Find a minimum sum-of-products realization for the expression f (a, b, c, d)
Part II
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Solution
Known Quantities: Logical expression; don t care conditions Find: Minimal realization Schematics, Diagrams, Circuits, and Given Data
cd ab 00 01 11 10 00 1 0 0 0 01 0 0 x 1 11 1 0 x x 10 0 1 x x
f (a, b, c, d) = 1 for {a, b, c, d} = {0000, 0011, 0110, 0101} f (a, b, c, d) = don t care for {a, b, c, d} = {1010, 1011, 1101, 1110, 1111}
Analysis: We cover the Karnaugh map of Figure 1354 using ones, and also using x
entries for each don t care condition By appropriately selecting three of the four don t care entries to be equal to 1, we complete the covering with one four-cell subcube, two two-cell subcubes, and one one-cell subcube, to obtain the following expression: f (a, b, c) = a b c d + b c d + a d + b c d
Comments: Would the product-of-sums realization be simpler Verify
Check Your Understanding
1315 Simplify the following expression to show that it corresponds to the function Z:
W X Y Z+W X Y Z+W X Y Z+W X Y Z+W X Y Z +W X Y Z + W X Y Z + W X Y Z
1316 Simplify the following expression, using a Karnaugh map:
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