Boolean identities: Which option correctly expresses the commutative law of multiplication for logical variables?

Introduction / Context:
Commutativity is a foundational property in Boolean algebra and ordinary algebra. It states that the order of operands in a binary operation does not affect the result. Recognizing commutative properties helps reorder and simplify expressions without changing their logical function.

Given Data / Assumptions:

• Boolean variables A, B, C represent logic 0/1.
• Operations: + is OR, • is AND.
• We seek the law for multiplication (AND).

Concept / Approach:
The commutative law for AND says A • B = B • A. Similarly, for OR: A + B = B + A. These laws allow rearrangement of terms, which often enables grouping or factoring that simplifies the expression further.

Step-by-Step Solution:
Identify operation: multiplication corresponds to logical AND.Apply commutativity: order of A and B is interchangeable for AND.Confirm by truth table: evaluate both sides for all combinations of A and B; outputs match.Use in simplification: reordering can expose common factors or enable distributive reductions.

Verification / Alternative check:
Karnaugh maps are unaffected by permutation of variable order; this reflects underlying commutativity. Algebraic proofs also follow directly from Boolean postulates.

Why Other Options Are Wrong:
A + B = B + A: commutative law of addition (OR), not multiplication.A • B = B + A: mixes AND and OR; not an identity.A • (B • C) = (A • B) • C: this is the associative law, not the commutative law.

Common Pitfalls:
Confusing commutative with associative properties; the former swaps operands, the latter changes grouping. Keep symbols consistent to avoid misapplication.

Final Answer:
A • B = B • A