Boolean algebra — identify the correct distributive law identity. Which of the following expressions correctly states the distributive law used in simplifying logic (algebra over {0,1})?

Introduction / Context:
The distributive law is one of the foundational identities in Boolean algebra, just as in ordinary arithmetic. It allows us to distribute multiplication over addition, or addition over multiplication, enabling systematic simplification of logic expressions and reduction of gate counts in digital circuits.

Given Data / Assumptions:

• We are working with Boolean variables A, B, C taking values in {0, 1}.
• Boolean '+' denotes logical OR, and adjacency (AB) denotes logical AND.
• We want the standard form of the distributive law that is most commonly applied in logic simplification.

Concept / Approach:
In Boolean algebra, both distributive identities hold: A(B + C) = AB + AC and A + BC = (A + B)(A + C). The first is the direct analog of arithmetic distribution of multiplication over addition and is used constantly to expand or factor expressions into sum-of-products or product-of-sums forms.

Step-by-Step Solution:

Verification / Alternative check:
Construct a small truth table for B + C and then AND with A. Compare its output bit-by-bit with AB + AC. You will observe identical outputs for all eight combinations of A, B, and C, confirming the equivalence.

Why Other Options Are Wrong:

• (A + B) + C = A + (B + C): this is associativity of OR, not distribution.
• A + (B + C) = AB + AC: mixes OR on the left with AND on the right; not a valid identity.
• A(BC) = (AB) + C: incorrectly replaces an AND with an OR.
• (A + B)C = AC + BC: this is actually another valid distributive identity; however, the classic statement asked for in many basics questions is A(B + C) = AB + AC. If multiple correct forms are permitted, both apply; here we select the canonical version provided in option A.

Common Pitfalls:

• Confusing association (grouping) with distribution (expanding across different operators).
• Accidentally changing operators during expansion (e.g., turning AND into OR).

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