The minimal expression of A + A'BA is
2026
The minimal expression of A + A'BA is
Answer: A. A — Concept: In Boolean algebra, ANDing a variable with its own complement always gives 0 (Complement Law: X . X' = 0), and ORing any expression with 0 leaves it…
- A.
A
- B.
A + B
- C.
AB
- D.
B
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Correct answer: A
Concept: In Boolean algebra, ANDing a variable with its own complement always gives 0 (Complement Law: X . X' = 0), and ORing any expression with 0 leaves it unchanged (Identity Law: X + 0 = X). The AND operation is commutative and associative, so the literals inside a product term can be regrouped in any order without changing its value.
Application:
Start with the given expression: A + A'BA.
Regroup the AND term using commutativity and associativity: A'BA = A . A' . B.
Apply the Complement Law to the regrouped pair: A . A' = 0, so A'BA = 0 . B = 0.
Substitute this back into the original expression: A + A'BA = A + 0.
Apply the Identity Law: A + 0 = A.
The expression reduces to the single literal A, which cannot be simplified further, so this is the minimal form.
Cross-check: Verifying with a truth table over every value of A and B confirms the reduction:
A | B | A' | A'BA | A + A'BA |
|---|---|---|---|---|
0 | 0 | 1 | 0 | 0 |
0 | 1 | 1 | 0 | 0 |
1 | 0 | 0 | 0 | 1 |
1 | 1 | 0 | 0 | 1 |
Every row of A + A'BA matches the value of A exactly, confirming that the minimal expression is A.