Digital Logic Toolkit — Boolean & Minimisation
Boolean algebra and minimisation
Boolean algebra is the arithmetic of two values, and every tool in this section is a different view of the same object: an expression, a table, a map, a list of rows. Enter a function in any one of them and the rest follow.
The seven tools
Boolean algebra simplifier
Simplify any expression to its minimal SOP and POS, one named law at a time.
Typical inputA·B + Ā·C + B·C →
Truth table generator
Turn an expression into a full truth table, with a column for every sub-expression.
Typical input(p → q) ∧ (q → r) →
Karnaugh map solver
Group a 2- to 6-variable map yourself and have every group marked right or wrong.
Typical inputΣm(1,3,7,11,15) + d(0,2,5) →
Two equally minimal answers. Most tools show one and imply it is the only one.
Quine–McCluskey solver
Minimise past the K-map limit with the full tabular method and Petrick’s step.
Typical inputΣm(0,1,2,5,6,7,8,9,10,13,14) →
Four equally minimal answers, and the tool lists all four.
Boolean laws & theorems
Every law, both duals, each with a truth-table proof you can check.
Typical inputA + Ā·B = A + B →
De Morgan’s laws
Push a negation through any expression and see both forms side by side.
Typical input(A·B·C)′ →
SOP & POS canonical forms
Expand to canonical minterms and maxterms, and convert between the two.
Typical inputΣm(1,3,6,7) →
The one thing worth knowing before you start
A Karnaugh map grouping is three algebra steps. Two cells that differ in one variable give t·x + t·x′; factor with the Distributive law to get t·(x + x′); the Complement law makes the bracket 1; the Identity law drops it, leaving t. That chain — usually named the Adjacency or Combining law — is why a group of two removes one literal, a group of four removes two, and a group of eight removes three.
How we know the answers are right
Minimisation runs Quine–McCluskey to a proven minimum before any explanation is written, so the working narrates a result that is already known to be optimal rather than stopping wherever the rewrite rules run out. Every emitted step is checked against the original function’s truth table in test builds. The full test suite, including the exhaustive check over all 65,536 functions of four variables, is published.
How the answers are computed, in full →
Related
- Gates & Circuits
What each gate does, in both symbol standards, and how to build circuits out of them.
- Numbers & Codes
Convert between bases and representations, and check the codes that detect and fix errors.
- Logic gate symbols
Every gate in both symbol standards, side by side, on a printable one-page chart.