The Boolean expression Y = (A + B̅ + A̅B)C̅ simplifies to:
2007
The Boolean expression Y = (A + B̅ + A̅B)C̅ simplifies to:
Answer: C. C̅ — ConceptBoolean expressions can be simplified with equivalence laws. The absorption law is X + X̅Y = X + Y; the complement law is X + X̅ = 1; and the identity…
- A.
A C̅
- B.
B C̅
- C.
C̅
- D.
AB
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Correct answer: C
Concept
Boolean expressions can be simplified with equivalence laws. The absorption law is X + X̅Y = X + Y; the complement law is X + X̅ = 1; and the identity law is 1·Z = Z.
Apply these laws to the bracketed factor before multiplying by the outer term.
Application
Regroup the bracket: A + B̅ + A̅B = (A + A̅B) + B̅.
Use absorption: A + A̅B = A + B.
Substitute it back: (A + B) + B̅ = A + (B + B̅).
Use the complement law and domination: A + (B + B̅) = A + 1 = 1.
Therefore Y = 1·C̅ = C̅.
Cross-check
The bracket is always 1, so Y must follow only the complement of C.
C | C̅ | Bracket | Y |
|---|---|---|---|
0 | 1 | 1 | 1 |
1 | 0 | 1 | 0 |
Thus the simplified value is C̅.