Discussion
In Boolean algebra, \( A \cdot (B + C) = A\cdot B + A\cdot C \).
Answer: A
Correct answer: True
The statement is true. This is the distributive law, and it is the same identity as in ordinary algebra, where multiplication distributes over addition. What has no ordinary counterpart is its dual, which also holds in Boolean algebra: \[ A + B\cdot C = (A + B)(A + C). \] Addition distributes over multiplication, which is false for numbers and true here - a good illustration that Boolean algebra is a different system despite the familiar notation. Every Boolean identity has such a dual, obtained by exchanging AND with OR and 0 with 1. The pairing runs through the whole set: \( A + 0 = A \) with \( A \cdot 1 = A \), \( A + 1 = 1 \) with \( A \cdot 0 = 0 \), and De Morgan’s two theorems with each other. The distributive law is what allows an expression to be converted between sum-of-products and product-of-sums form, which is the first step in most simplification.
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