Boolean algebra is the branch of mathematics that describes logical relationships between binary variables, forming the mathematical foundation of every digital circuit ever built. Developed by George Boole in 1854 and connected to switching circuits by Claude Shannon in 1938, it allows designers to express, analyze, and minimize any logic circuit using just two values — 0 (false) and 1 (true). The key insight every practitioner must internalize is the Duality Principle: every valid Boolean identity produces a second valid identity by swapping AND↔OR and 0↔1, which effectively doubles the utility of every rule learned and explains why the laws always appear in pairs.
What This Cheat Sheet Covers
This topic spans 12 focused tables and 96 indexed concepts. Below is a complete table-by-table outline of this topic, spanning foundational concepts through advanced details.
Table 1: Fundamentals — Variables, Notation, and Operator Precedence
Boolean expressions are built from a small, precise vocabulary; knowing the exact meaning of each term prevents the most common confusion in logic simplification and circuit analysis.
| Concept | Example | Description |
|---|---|---|
A, B, X, Y | A named symbol that holds exactly one value: 0 or 1. | |
A, A', B, B' | • A variable or its complement as it appears in an expression • every literal is either positive ( A) or negative (A'). | |
A' (also \overline{A} or !A) | • The logical inverse of a variable • if A = 1 then A' = 0, and vice versa | |
A·B·C' | • AND of literals • called a minterm when it contains every variable exactly once | |
A + B' + C | • OR of literals • called a maxterm when it contains every variable exactly once | |
F = AB + A'C | • A combination of literals with AND/OR/NOT operators • maps any set of variable values to 0 or 1. |