Meaning of the commutative law in Boolean algebra What does the commutative property of Boolean addition and multiplication imply about the ordering of variables in two-input OR and AND operations?
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Awe can group variables in an AND or in an OR any way we want
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Ban expression can be expanded by multiplying term by term just the same as in ordinary algebra
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Cthe way we OR or AND two variables is unimportant because the result is the same
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Dthe factoring of Boolean expressions requires the multiplication of product terms that contain like variables
Answer
Correct Answer: the way we OR or AND two variables is unimportant because the result is the same
Explanation
Introduction / Context:The commutative law is a foundational identity in Boolean algebra and arithmetic. In logic design, it allows flexibility in wiring and simplifies reasoning: swapping the order of inputs to an AND or OR gate does not change the output.
Given Data / Assumptions:
- Binary variables A, B representing logical 0/1.
- Operations: OR (+) and AND (·).
- We focus on two-input operations for clarity.
Concept / Approach:Commutative law states: A + B = B + A and A * B = B * A. Therefore, input order to OR or AND gates is irrelevant. This is distinct from the associative law (regrouping more than two terms) and the distributive law (mixing AND/OR with distribution).
Step-by-Step Solution:
State the identities: A + B = B + A; A * B = B * A.Interpretation: swapping inputs does not alter the result.Practical implication: wiring order into a gate can be chosen for layout convenience.Verification / Alternative check:Truth tables confirm equality for all 4 input combinations (00, 01, 10, 11). Outputs match irrespective of input order for both OR and AND operations.
Why Other Options Are Wrong:
- Grouping “any way we want” (Option A) describes associativity, not commutativity.
- Term-by-term expansion (Option B) references distributivity.
- Factoring requirement (Option D) is unrelated to commutativity.
Common Pitfalls:Confusing commutative with associative laws; believing gate pin numbering affects logic (it does not for symmetric gates like AND/OR). Non-commutative logic operations exist in other algebras, but not for standard Boolean AND/OR.
Final Answer:the way we OR or AND two variables is unimportant because the result is the same