Boolean algebra fundamentals: Which one of the following identities correctly expresses the distributive law in Boolean algebra?

Introduction / Context:
In digital logic, Boolean algebra provides the rule set used to manipulate logic expressions before implementing them with gates. The distributive law is one of the most used identities when converting between sum-of-products and product-of-sums forms or when factoring and expanding logic to reduce gate count.

Given Data / Assumptions:

• Standard Boolean operators: + means OR, juxtaposition or · means AND, and the complement is shown by a bar or apostrophe.
• We seek the correct statement of the distributive law (multiplication over addition or addition over multiplication) in Boolean algebra.

Concept / Approach:
The Boolean distributive law mirrors arithmetic in structure but with OR and AND. The most common form is A(B + C) = AB + AC, which states that AND distributes over OR. There is also a dual form where OR distributes over AND: A + BC = (A + B)(A + C). Recognizing these forms allows fast algebraic simplification.

Step-by-Step Solution:
Identify candidate that shows AND distributing across OR.Check option: A(B + C) on the left should expand to AB + AC on the right.Result matches the standard identity → select A(B + C) = AB + AC.

Verification / Alternative check:
Construct a two-level truth check: For each combination of A, B, C, evaluate left side A(B + C) and right side AB + AC. They match for all 8 combinations, confirming the identity. Alternatively, use factoring in reverse: AB + AC = A(B + C).

Why Other Options Are Wrong:
(A + B) + C = A + (B + C): This is associativity of OR, not distributivity.A + (B + C) = AB + AC: Left is pure OR, right is AND/OR mix; not an identity.A(BC) = (AB) + C: No standard identity supports this; left is AND-only, right introduces OR improperly.(A + B)C = AC + BC: Although true, it is a different expression than option B; it illustrates C(A + B) form. If present, it also represents distributivity (OR distributing over multiplication after factoring C). It is correct but not the classic base form being asked; option B is the canonical textbook statement.

Common Pitfalls:
Confusing associativity with distributivity or mixing arithmetic intuition without checking Boolean operator meanings.

Final Answer:
A(B + C) = AB + AC