Consider the Boolean function used in a CPU control unit: Which of the…
2025
Consider the Boolean function used in a CPU control unit:
Which of the following expressions is the simplified form of Y?
Attempted by 15 students.
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Concept: A product of two Boolean sums is expanded by the distributive law: every term of the first sum is ANDed with every term of the second. Two identities then prune the expansion — the complement law X · X̄ = 0, which annihilates any product that contains a variable together with its own complement, and the idempotent law X · X = X, which collapses a repeated literal.
Application: Distributing the two terms of (A + B·C̄) over the two terms of (Ā·B + C) produces four products:
A · (Ā·B) = (A·Ā)·B = 0·B = 0, because A and Ā can never be 1 together.
A · C = A·C, which contains no complementary pair and therefore survives.
(B·C̄) · (Ā·B) = Ā·(B·B)·C̄ = Ā·B·C̄, where the repeated B collapses by idempotence.
(B·C̄) · C = B·(C̄·C) = B·0 = 0, because C and C̄ can never be 1 together.
Adding the four products and dropping the two zero terms gives:
Y = 0 + A·C + Ā·B·C̄ + 0 = A·C + Ā·B·C̄
Cross-check: Evaluate the original function and the reduced expression on all eight input combinations; they agree on every row, so the reduction is exact.
A | B | C | Y = (A + B·C̄)·(Ā·B + C) | A·C + Ā·B·C̄ |
|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 1 | 0 | 1 | 1 |
0 | 1 | 1 | 0 | 0 |
1 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 1 | 1 |
1 | 1 | 0 | 0 | 0 |
1 | 1 | 1 | 1 | 1 |
Contrast: The other listed expressions each disagree with Y on at least one row:
A·B + C outputs 1 at A = 0, B = 0, C = 1, where Y is 0.
A·C + B·C̄ outputs 1 at A = 1, B = 1, C = 0, where Y is 0.
A·C + Ā·B + B·C̄ outputs 1 at A = 0, B = 1, C = 1, where Y is 0.
Hence the simplified form of the control-unit function is Y = A·C + Ā·B·C̄.