Consider the following Boolean function in Sum of Products form: F(x, y, z) =…
Consider the following Boolean function in Sum of Products form:
F(x, y, z) = xyz' + xy'z' + x'yz + x'y'z'
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Solution:
Write the original sum-of-products and identify the minterms.
xyz' corresponds to minterm 110 (decimal 6).
xy'z' corresponds to minterm 100 (decimal 4).
x' y z corresponds to minterm 011 (decimal 3).
x' y' z' corresponds to minterm 000 (decimal 0).
Combine obvious pairs to simplify.
xyz' + xy'z' = x z' (factor x z' out because y + y' = 1).
x'y'z' and xy'z'? The pair of minterms 000 and 100 (decimals 0 and 4) share y' z', giving y' z' as a prime implicant that covers minterms 0 and 4.
Include any remaining required implicants.
The minterm 011 (decimal 3) is only covered by x' y z, so this implicant must be included.
Assemble the minimal cover.
The minimal sum-of-products that covers all minterms {0,3,4,6} is:
Final simplified expression: x z' + y' z' + x' y z