The simplified form of the Boolean equation \((A\bar{B}+A\bar{B}+AC)(\bar{A}\ba…
2016
The simplified form of the Boolean equation \((A\bar{B}+A\bar{B}+AC)(\bar{A}\bar{C}+\bar{B})\) is
- A.
\(A\bar{B}\) - B.
\(A\bar{B}C\) - C.
\(\bar{A}B\) - D.
\(ABC\)
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Correct answer: A
Simplify (A\bar{B}+A\bar{B}+AC)(\bar{A}\bar{C}+\bar{B}):
Remove the duplicate term: A\bar{B}+A\bar{B}+AC = A\bar{B}+AC.
Factor A: A\bar{B}+AC = A(\bar{B}+C). So the whole expression becomes A(\bar{B}+C)(\bar{A}\bar{C}+\bar{B}).
Use the identity (p+q)(p+r) = p + q r with p = \bar{B}, q = C, r = \bar{A}\bar{C}:
Therefore (\bar{B}+C)(\bar{B}+\bar{A}\bar{C}) = \bar{B} + C·\bar{A}\bar{C}.
But C·\bar{C} = 0, so C·\bar{A}\bar{C} = 0, leaving (\bar{B}+C)(\bar{A}\bar{C}+\bar{B}) = \bar{B}.
Multiply by the leading A: A·\bar{B}.
Final answer: A\bar{B}