Boolean Algebra and Logic Gates Cheat Sheet

Updated 2026-05-19

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 PrecedenceTable 2: Core Logic Gates — AND, OR, NOT, BufferTable 3: Derived Logic Gates — NAND, NOR, XOR, XNORTable 4: Boolean Algebra LawsTable 5: De Morgan's TheoremsTable 6: Advanced Theorems — Duality, Consensus, Shannon Expansion, TranspositionTable 7: Canonical and Standard Forms — Minterms, Maxterms, SOP, POSTable 8: Karnaugh Map (K-map) MinimizationTable 9: K-map Advanced — Prime Implicants and Don't-Care ConditionsTable 10: Universal Gates — Implementing Any Function with NAND or NORTable 11: Quine-McCluskey Tabular MinimizationTable 12: Boolean Algebra Applications in Digital Circuits

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
Variable	`A`, `B`, `X`, `Y`	A named symbol that holds exactly one value: 0 or 1.
Literal	`A`, `A'`, `B`, `B'`	A variable or its complement as it appears in an expression; every literal is either positive (`A`) or negative (`A'`).
Complement	`A'` (also $\overline{A}$ or `!A`)	The logical inverse of a variable; if A = 1 then A' = 0, and vice versa.
Product term	`A·B·C'`	AND of literals; called a minterm when it contains every variable exactly once.
Sum term	`A + B' + C`	OR of literals; called a maxterm when it contains every variable exactly once.
Boolean expression	`F = AB + A'C`	A combination of literals with AND/OR/NOT operators; maps any set of variable values to 0 or 1.

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
Variable	`A`, `B`, `X`, `Y`	A named symbol that holds exactly one value: 0 or 1.
Literal	`A`, `A'`, `B`, `B'`	A variable or its complement as it appears in an expression; every literal is either positive (`A`) or negative (`A'`).
Complement	`A'` (also $\overline{A}$ or `!A`)	The logical inverse of a variable; if A = 1 then A' = 0, and vice versa.
Product term	`A·B·C'`	AND of literals; called a minterm when it contains every variable exactly once.
Sum term	`A + B' + C`	OR of literals; called a maxterm when it contains every variable exactly once.
Boolean expression	`F = AB + A'C`	A combination of literals with AND/OR/NOT operators; maps any set of variable values to 0 or 1.