Which of the following Boolean algebraic equation(s) is/are CORRECT?
2025
Which of the following Boolean algebraic equation(s) is/are CORRECT?
- A.
\(\overline{A}BC + A\overline{B}\,\overline{C} + \overline{A}\,\overline{B}\,\overline{C} + A\overline{B}C + ABC = BC + \overline{B}\,\overline{C} + \overline{A} \overline{B}\) - B.
\(AB + \overline{A}C + BC = AB + \overline{A}C\) - C.
\((A + C)(\overline{A} + B) = AB + \overline{A}C\) - D.
\(\overline{(A + \overline{B} + \overline{D})(C + D)(\overline{A} + C + D)(A + B + \overline{D})} = \overline{A}D + \overline{C} \overline{D}\)
Attempted by 103 students.
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Correct answer: B, C, D
Summary of which equalities hold:
The expression A'BC + AB'C' + A'B'C' + AB'C + ABC simplifies to AC + BC + B'C', so the provided right-hand side BC + B'C' + A'B is not equivalent.
AB + A' C + BC simplifies to AB + A' C (BC is a redundant consensus term), so the equality holds.
(A + C)(A' + B) expands to AB + A' C + BC, which reduces to AB + A' C, so the equality holds.
The complement of (A + B' + D')(C + D)(A' + C + D)(A + B + D') simplifies via De Morgan and reduction to A' D + C' D', so the equality holds.
Detailed derivations:
Simplify A'BC + AB'C' + A'B'C' + AB'C + ABC:
Group terms with C: C(A'B + AB' + AB) = C(A + B) = AC + BC. The remaining terms are B'C'. Therefore the whole expression equals AC + BC + B'C'.
Conclusion: the stated right-hand side BC + B'C' + A'B is not equivalent; a counterexample is A=1, B=0, C=1, which gives left = 1 but the stated right-hand side = 0.
Simplify AB + A' C + BC:
Use the consensus theorem: XY + X'Z + YZ = XY + X'Z. Here X=A, Y=B, Z=C. So AB + A' C + BC = AB + A' C.
Simplify (A + C)(A' + B):
Expand: (A + C)(A' + B) = AA' + AB + CA' + CB = AB + A' C + BC. Apply consensus to drop BC and obtain AB + A' C, matching the right-hand side.
Simplify the complement expression:
Apply De Morgan: the complement of a product is the OR of the complements of each factor. The factor complements are:
(A + B' + D')' = A' B D
(C + D)' = C' D'
(A' + C + D)' = A C' D'
(A + B + D')' = A' B' D
OR these terms: A' B D + C' D' + A C' D' + A' B' D. Combine A' B D + A' B' D = A' D, and note C' D' already covers A C' D'. The result is A' D + C' D', matching the stated right-hand side.
Final correct equalities (the expressions that are valid equalities):
AB + A' C + BC = AB + A' C
(A + C)(A' + B) = AB + A' C
( (A + B' + D')(C + D)(A' + C + D)(A + B + D') )' = A' D + C' D'
Note: The originally marked correct option (the long five-term sum equated to BC + B'C' + A'B) is incorrect as shown above.
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