Dual of a Boolean theorem is obtained by
2013
Dual of a Boolean theorem is obtained by
- A.
Interchanging all zero and ones only
- B.
Changing all zero to one only
- C.
Changing all ones to zero
- D.
Interchanging all zero and ones and . to +
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Correct answer: D
Concept
In Boolean algebra, the dual of a statement is formed by two simultaneous substitutions: every AND operator (·) is swapped with OR (+) and vice versa, and every constant 0 is swapped with 1 and vice versa. Variables and their complements are left unchanged. The Principle of Duality guarantees that if an identity is valid, its dual is automatically valid too.
Application
Applying the rule to a Boolean theorem requires both swaps together:
Interchange the operators: each · (AND) becomes + (OR) and each + becomes ·.
Interchange the constants: each 0 becomes 1 and each 1 becomes 0.
Keep all variables exactly as they are.
The correct response is therefore the one that combines an interchange of the constants 0 and 1 with a change between the · and + operators, rather than a substitution that only touches the constants.
Cross-check
Take the identity A + 0 = A. Applying both swaps gives A · 1 = A, which is also a valid identity — confirming the dual needs both substitutions. If only the constants were swapped while the operators stayed put, A + 0 = A would become A + 1 = A, which is false; that is why an operator-only or constant-only transformation cannot be the dual.