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?

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:

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