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Karnaugh Karnaugh Map Method

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Karnaugh Map Technique K-Maps, like truth tables, are a way to show the relationship between logic inputs and desired outputs. K-Maps are a graphical technique used to simplify a logic equation. K-Maps are very procedural and much cleaner than Boolean simplification. K-Maps can be used for any number of input variables, BUT are only practical for fewer than six.

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K-Map Format Each minterm in a truth table corresponds to a cell in the K-Map. K-Map cells are labeled so that both horizontal and vertical movement differ only in one variable. Once a K-Map is filled (0’s & 1’s) the sum-of- products expression for the function can be obtained by OR-ing together the cells that contain 1’s. Since the adjacent cells differ by only one variable, they can be grouped to create simpler terms in the sum-of-product expression.

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Y Y XX 0 1 2 3 Truth Table -TO- K-Map Y0101Y0101 Z1011Z1011 X0011X0011 minterm 0 minterm 1 minterm 2 minterm 3 1 1 0 1

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Y Y XX 0 0 1 0 X Y Y Y XX 0 0 0 1 Y Y XX 1 0 0 0 Y Y XX 0 1 0 0 2 Variable K-Map : Groups of One

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Adjacent Cells X Y Y Y XX 1 0 1 0 Y Y XX 1 0 1 0 Y = Z Z = X Y + X Y = Y ( X + X ) = Y 1

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Groupings Grouping a pair of adjacent 1’s eliminates the variable that appears in complemented and uncomplemented form. Grouping a quad of 1’s eliminates the two variables that appear in both complemented and uncomplemented form. Grouping an octet of 1’s eliminates the three variables that appear in both complemented and uncomplemented form, etc…..

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Y Y XX 1 1 0 0 X X Y Y XX 1 0 1 0 Y Y 2 Variable K-Map : Groups of Two Y Y XX 0 1 0 1 Y Y XX 0 0 1 1

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Y Y XX 1 1 1 1 1 2 Variable K-Map : Group of Four

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Two Variable Design Example S S RR 0 1 2 3 S0101S0101 T1010T1010 R0011R0011 1 0 1 0 S T = F (R,S) = S

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3 Variable K-Map : Vertical minterm 0 minterm 1 minterm 2 minterm 3 minterm 4 minterm 5 minterm 6 minterm 7 C01010101C01010101 Y10110010Y10110010 B00110011B00110011 A00001111A00001111 1 0 0 0 1 1 0 1 AA B C 0 1 4 5 3 2 7 6

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3 Variable K-Map : Horizontal C C A B minterm 0 minterm 1 minterm 2 minterm 3 minterm 4 minterm 5 minterm 6 minterm 7 C01010101C01010101 Y10110010Y10110010 B00110011B00110011 A00001111A00001111 1 0 1 1 1 0 0 0 0 1 2 3 6 7 4 5

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3 Variable K-Map : Groups of Two C C A B 1 0 1 0 0 0 0 0 A C 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0 B C 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 A B 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1

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3 Variable K-Map : Groups of Four C C A B 1 1 1 1 0 0 0 0 A 0 0 0 0 1 1 1 1 A 0 0 1 1 1 1 0 0 B 1 1 0 0 0 0 1 1 B 1 0 1 0 1 0 1 0 C 0 1 0 1 0 1 0 1 C

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3 Variable K-Map : Group of Eight C C A B 1 1 1 1 1 1 1 1 1

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Simplification Process 1. Construct the K-Map and place 1’s in cells corresponding to the 1’s in the truth table. Place 0’s in the other cells. 2. Examine the map for adjacent 1’s and group those 1’s which are NOT adjacent to any others. These are called isolated 1’s. 3. Group any hex. 4. Group any octet, even if it contains some 1’s already grouped, but are not enclosed in a hex. 5. Group any quad, even if it contains some 1’s already grouped, but are not enclosed in a hex or octet. 6. Group any pair, even if it contains some 1’s already grouped, but are not enclosed in a hex, octet or quad. 7. Group any single cells remaining. 8. Form the OR sum of all the terms grouped.

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Three Variable Design Example #1 L01010101L01010101 M10110100M10110100 K00110011K00110011 J00001111J00001111 1 0 1 1 0 0 0 1 L L J K 0 1 2 3 6 7 4 5 J L J K J K L M = F (J,K,L) = J L + J K + J K L

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Three Variable Design Example #2 C01010101C01010101 Z10001101Z10001101 B00110011B00110011 A00001111A00001111 1 0 0 0 0 1 1 1 C C A B 0 1 2 3 6 7 4 5 B C A C Z = F (A,B,C) = A C + B C

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Three Variable Design Example #3 C01010101C01010101 F2 1 0 1 0 1 B00110011B00110011 A00001111A00001111 1 1 0 1 1 1 0 0 A A B C 01 2 3 6 7 4 5 A B A C F 2 = F (A,B,C) = B C + B C + A B F 2 = F (A,B,C) = B C + B C + A C

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Four Variable K-Map minterm 0 minterm 1 minterm 2 minterm 3 minterm 4 minterm 5 minterm 6 minterm 7 minterm 8 minterm 9 minterm 10 minterm 11 minterm 12 minterm 13 minterm 14 minterm 15 Z0101010101010101Z0101010101010101 F1 1 0 1 0 1 0 1 Y0011001100110011Y0011001100110011 X0000111100001111X0000111100001111 W0000000011111111W0000000011111111 0 1 4 5 12 13 8 9 3 2 7 6 15 14 11 10 W X Y Z 0 0 1 0 1 1 0 0 1 0 1 1 0 1 1 1

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Four Variable K-Map : Groups of Four W X Y Z 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 X Z 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0

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Four Variable Design Example #1 Z0101010101010101Z0101010101010101 F1 1 0 1 0 1 0 1 0 1 0 1 0 Y0011001100110011Y0011001100110011 X0000111100001111X0000111100001111 W0000000011111111W0000000011111111 0 1 4 5 12 13 8 9 3 2 7 6 15 14 11 10 W X Y Z 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 W X Y X Y Z W Z F1 = F (w,x,y,z) = W X Y + W Z + X Y Z min 0 min 15

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Four Variable Design Example #2 Z0101010101010101Z0101010101010101 F2 1 x 1 0 x 0 x 1 0 1 x 1 Y0011001100110011Y0011001100110011 X0000111100001111X0000111100001111 W0000000011111111W0000000011111111 0 1 4 5 12 13 8 9 3 2 7 6 15 14 11 10 W X Y Z X X 1 1 1 1 1 0 1 0 X X 0 X 1 0 F2 = F (w,x,y,z) = X Y Z + Y Z + X Y X Y Z X Y min 0 min 15

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