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05-AdditionalKmaps - Example K-map simplification Let's...

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CS231 Boolean Algebra 1 Example K-map simplification Let’s consider simplifying f(x,y,z) = xy + y’z + xz . First, you should convert theexpression into a sum of minterms form, if it’s not already. Theeasiest way to do this is to makea truth tablefor thefunction, and then read off the minterms. You can either writeout theliterals or usetheminterm shorthand. Hereis thetruth tableand sum of minterms for our example: x y z f (x,y,z) 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1 f(x,y,z) = x’y’z + xy’z + xyz’ + xyz
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CS231 Boolean Algebra 2 Unsimplifying expressions You can also convert theexpression to a sum of minterms with Boolean algebra. Apply thedistributivelaw in reverseto add in missing variables. Very few peopleactually do this, but it’s occasionally useful. In both cases, we’reactually “unsimplifying” our exampleexpression. Theresulting expression is larger than theoriginal one! But having all theindividual minterms makes it easy to combinethem together with theK- map. xy + y’z + xz = (xy 1) + (y’z 1) + (xz 1) = (xy (z’ + z)) + (y’z (x’ + x)) + (xz (y’ + y)) = (xyz’ + xyz) + (x’y’z + xy’z) + (xy’z + xyz) = xyz’ + xyz + x’y’z + xy’z
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CS231 Boolean Algebra 3 Making the example K-map Next up is drawing and filling in theK-map. Put 1s in themap for each minterm, and 0s in theother squares. You can useeither themintermproducts or theshorthand to show you wherethe1s and 0s belong. In our example, wecan writef(x,y,z) in two equivalent ways. In either case, theresulting K-map is shown below. Y 0 1 0 0 X 0 1 1 1 Z Y x y z x y z x yz x yz X xy z xy z xyz xyz Z f(x,y,z) = x’y’z + xy’z + xyz’ + xyz Y m 0 m 1 m 3 m 2 X m 4 m 5 m 7 m 6 Z
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CS231 Boolean Algebra 4 K-maps from truth tables You can also fill in theK-map directly from a truth table. Theoutput in row i of thetablegoes into squarem i of theK-map. Remember that therightmost columns of theK-map are“switched.” Y m 0 m 1 m 3 m 2 X m 4 m 5 m 7 m 6 Z x y z f (x,y,z) 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1 Y 0 1 0 0 X 0 1 1 1 Z
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CS231 Boolean Algebra 5 Grouping the minterms together Themost difficult step is grouping together all the1s in theK-map. Make rectangles around groups of one, two, four or eight 1s.
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