Discussion
Which of the following options represents the simplified form of the Boolean equation \(Y = \bar{A}\bar{B}\bar{C} + \bar{A}B\bar{C} + A\bar{B}\bar{C} + AB\bar{C}\) ?
Answer: D
The expression simplifies to \(\bar{C}\).
Every term contains \(\bar{C}\), so it can be taken out as a common factor. What is left is the sum of all four combinations of \(A\) and \(B\), which is always 1:
\[ \begin{aligned} Y &= \bar{C}(\bar{A}\bar{B} + \bar{A}B + A\bar{B} + AB) \\ &= \bar{C}\,[\bar{A}(\bar{B} + B) + A(\bar{B} + B)] \\ &= \bar{C}(\bar{A} + A) = \bar{C} \end{aligned} \]
So \(Y\) is 1 whenever \(C = 0\), whatever \(A\) and \(B\) are. \(A\bar{B}\) and \(\bar{A}B\bar{C}\) cover only one or two of the four minterms, and \(\bar{B}C\) requires \(C = 1\), where every term of \(Y\) is 0.
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