Meaning of the commutative laws: What do the commutative laws of addition (OR) and multiplication (AND) indicate about the order of operands in Boolean expressions?

Introduction / Context:
Commutative laws are core algebraic properties that carry over to Boolean algebra. They empower designers to rearrange terms without changing the logic function, enabling cleaner factoring, easier Karnaugh mapping, and standardized gate ordering in schematics.

Given Data / Assumptions:

• Commutative law for OR: A + B = B + A.
• Commutative law for AND: A • B = B • A.
• Variables represent binary logic values.

Concept / Approach:
Order independence simplifies manipulation: whether A precedes B or vice versa, the value of the expression is unchanged. This is distinct from associative laws (which change grouping) and distributive laws (which change structure via factoring or expansion).

Step-by-Step Solution:
Identify operation type (OR or AND) and the operands.Swap operand order and observe unchanged truth table outputs.Apply this property to reorganize terms to reveal common factors or to line up literals for minimization.Use in practice during algebraic reductions and when labeling gate inputs; hardware is indifferent to the labeling order.

Verification / Alternative check:
Truth tables for A op B versus B op A (op ∈ {+, •}) are identical for all four input pairs. Tool-based equivalence checkers confirm commutativity automatically during logic synthesis.

Why Other Options Are Wrong:
Grouping any way we want: that is associativity, not commutativity.Term-by-term multiplication: describes distribution/expansion, not commutativity.Factoring requirement statement: unrelated and misleading.

Common Pitfalls:
Conflating commutative and associative properties; reordering and regrouping are different operations with separate laws.

Final Answer:
the way we OR or AND two variables is unimportant because the result is the same