Boolean Algebra — Associative Law (Addition/OR) Which expression correctly states the associative law for Boolean addition (logical OR), showing that grouping does not change the result?

Introduction / Context:
The associative law is one of the cornerstone identities in Boolean algebra. It states that when combining three or more variables with the same operator, the way they are grouped does not affect the final outcome. For OR (Boolean addition), this greatly simplifies expression manipulation and logic reduction.

Given Data / Assumptions:

• Operator “+” represents logical OR.
• Variables A, B, C take binary values 0 or 1.
• Standard Boolean algebra axioms apply.

Concept / Approach:
The associative law for OR says (A + B) + C = A + (B + C). This means parentheses can be moved without changing the result. The same property holds for AND: (AB)C = A(BC). Recognizing these equivalences helps restructure expressions and match available hardware gates.

Step-by-Step Solution:
1) Evaluate X1 = A + (B + C). If any of A, B, or C is 1, X1 is 1.2) Evaluate X2 = (A + B) + C. Again, if any of A, B, or C is 1, X2 is 1.3) Since X1 and X2 are identical for all eight input combinations, the expressions are equivalent.4) Therefore, A + (B + C) = (A + B) + C is the correct associative-law statement for OR.

Verification / Alternative check:
Create a truth table with all combinations of A, B, C. The columns for A + (B + C) and (A + B) + C match row by row, proving associativity.

Why Other Options Are Wrong:

• A + (B + C) = A + (BC): Mixes OR and AND; not an associative form.
• A(BC) = (AB) + C: Left is all-AND, right introduces OR; not equivalent.
• ABC = A + B + C: AND of all three is not equal to their OR; only true in trivial specific cases.

Common Pitfalls:
Confusing associativity (moving parentheses) with distributivity (mixing operators), and commutativity (swapping order).

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