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{{title|Karnaugh Map (K-map)}} | {{title|Karnaugh Map (K-map)}} | ||
− | <div style="float: right; text-align: center; margin: 20px; width: 250px">[[File:3-input MAJ gate kmap.svg|200px]]<br />3-input [[MAJ]] gate<br /><math> | + | <div style="float: right; text-align: center; margin: 20px; width: 250px"> |
+ | [[File:3-input MAJ gate kmap.svg|200px]]<br /> | ||
+ | 3-input [[MAJ]] gate<br /> | ||
+ | <math> | ||
\begin{align} | \begin{align} | ||
f(a,b,c) =& AB+AC+BC \\ | f(a,b,c) =& AB+AC+BC \\ | ||
=& \sum m(3,5,6,7) \\ | =& \sum m(3,5,6,7) \\ | ||
− | f(a,b,c) =& \prod M(0,1,2,4) | + | f^\prime(a,b,c) =& \prod M(0,1,2,4) |
\end{align} | \end{align} | ||
</math> | </math> | ||
</div> | </div> | ||
− | ''' | + | '''Marnaugh Map''' ('''K-map''') (pronounced ''car-no map'') is a graphical tool that provides a simple and straightforward method of [[logic minimization|minimizing]] [[Boolean algebra|Boolean expressions]]. The K-map method was introduced in 1953 by [[Maurice Karnaugh]] as an enhancement to [[Veitch diagram]]. |
== Map Construction == | == Map Construction == | ||
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\end{align} | \end{align} | ||
</math> | </math> | ||
− | Each minterm in the equation is | + | Each minterm in the equation is than transferred into the K-map where each variable in the minterm represents a 1 and each complemented variable represents a 0. |
[[File:kmap example color coded (expression).svg|400px]] | [[File:kmap example color coded (expression).svg|400px]] | ||
=== from truth table === | === from truth table === | ||
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[[File:kmap example 2.svg|right|100px]] | [[File:kmap example 2.svg|right|100px]] | ||
− | The Karnaugh map on the | + | The Karnaugh map on the left on the other hand has two groups. One group spans both <math>B = 0</math> and <math>B = 1</math> and another that spans <math>A = 0</math> and <math>A = 1</math>. In the expression for the group that spans vertically, <math>B</math> changes yielding the expression <math>A</math>. Likewise in the expression that spans horizontally, <math>A</math> changes, yielding the expression <math>B</math>. The simplified equation for this K-map is the ORing of all the individual term - <math>f(A,B) = \sum m(1,2,3) = A+B</math>. |
[[File:kmap example 3.svg|left|200px]] | [[File:kmap example 3.svg|left|200px]] | ||
In this K-map, cells 0 and 4 are considered adjacent as well as cells 3 and 7. For the group involving cells 0 and 4, <math>A</math> changes, therefore it is dropped from the expression. Because <math>B</math> is always 0 and <math>C</math> is always 0 as well, the equation for that group is <math>\bar B \bar C</math>. For the second group involving cells 3 and 7, <math>A</math> changes once again. In this group <math>B</math> is always 1 and <math>C</math> is always 1 as well. The equation for this group is <math>BC</math>. The final simplified equation for this K-map is the ORing of all the terms - <math>f(A,B,C) = \sum m(0,3,4,7) = \bar B \bar C + BC</math>. | In this K-map, cells 0 and 4 are considered adjacent as well as cells 3 and 7. For the group involving cells 0 and 4, <math>A</math> changes, therefore it is dropped from the expression. Because <math>B</math> is always 0 and <math>C</math> is always 0 as well, the equation for that group is <math>\bar B \bar C</math>. For the second group involving cells 3 and 7, <math>A</math> changes once again. In this group <math>B</math> is always 1 and <math>C</math> is always 1 as well. The equation for this group is <math>BC</math>. The final simplified equation for this K-map is the ORing of all the terms - <math>f(A,B,C) = \sum m(0,3,4,7) = \bar B \bar C + BC</math>. | ||
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