The dual of a Boolean expression is obtained by interchanging:
2013
The dual of a Boolean expression is obtained by interchanging:
- A.
the Boolean Sum (OR, +) and Boolean Product (AND, ·) operators only, keeping the constants 0's and 1's unchanged
- B.
the Boolean Sum and Boolean Product operators, or alternatively the constants 0's and 1's — either one substitution by itself
- C.
the Boolean Sum and Boolean Product operators and the constants 0's and 1's simultaneously, while every variable stays unchanged
- D.
only the constants 0's and 1's, keeping the operators unchanged
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Correct answer: C
Concept: The principle of duality in Boolean algebra states that for any valid Boolean identity, a second valid identity (its dual) can be produced by simultaneously (1) interchanging the OR (+) and AND (·) operators, and (2) interchanging the constants 0 and 1 — every variable and its complement are left unchanged. Both substitutions must be made together; making only one of the two generally breaks the identity.
Application: Apply the rule to any given Boolean expression by:
Replacing every Boolean Sum (+) operator with a Boolean Product (·) operator, and every Boolean Product (·) operator with a Boolean Sum (+) operator.
Replacing every constant 0 with 1 and every constant 1 with 0, in the same pass.
Leaving all variables (and their complements, if any) unchanged.
Cross-check: Check the rule on the identity A + 1 = 1. Applying BOTH substitutions together (+ -> ·, 1 -> 0) gives A · 0 = 0, which is itself a true Boolean identity — the operator swap and the constant swap were needed together to land on another valid identity. This paired operator+constant interchange is also what underlies De Morgan's theorems, (A + B)' = A' · B' and (A · B)' = A' + B'.
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