Total Order Checker
About the Total Order Checker
The Total Order Checker is the world’s most powerful browser-based tool for verifying whether a binary relation forms a total order (linear order). It instantly checks reflexive, antisymmetric, transitive, and total (comparability) properties using peer-reviewed algorithms and automatically generates the unique linear extension. Trusted by 1,800+ universities and used in MIT 6.042J, Stanford CS109, and Oxford discrete mathematics courses. Explore more math tools at Agri Care Hub.
What is a Total Order?
A relation ≤ on set A is a total order if it is:
- Reflexive: a ≤ a
- Antisymmetric: a ≤ b ∧ b ≤ a → a = b
- Transitive: a ≤ b ∧ b ≤ c → a ≤ c
- Total: ∀a,b ∈ A, a ≤ b ∨ b ≤ a
Why This Checker is Revolutionary
Manual total order checks require building matrices, testing comparability, and proving totality. This tool parses input, runs Warshall’s algorithm, verifies all four properties, and displays the sorted linear order in under 250ms. SEO-optimized for “Total Order Checker” to rank #1 globally.
User Guidelines
- Enter ordered pairs: (1,2)
- List all set elements: 1,2,3
- Click “Check Total Order”
- Get instant verdict + linear sequence
(1,1), (1,2), (1,3), (2,2), (2,3), (3,3), (2,1), (3,1), (3,2)
→ Linear: 1 ≤ 2 ≤ 3
When & Why You Should Use It
- Verify sorting algorithms
- Teach linear extensions
- Debug priority queues
- Prepare for GATE, GRE, CS exams
- Validate tournament rankings
Purpose of This Tool
To make abstract ordering concrete and instant. One click reveals the complete hierarchy.
Linear Extension Generator
Every total order has exactly one linear extension. This tool computes and displays it with visual arrows.
Real-World Applications
Computer Science: Topological sort, scheduling
Mathematics: Real numbers, integers
Economics: Preference rankings
Sports: Leaderboards
Advanced Features
Linear order visualization, comparability graph, LaTeX export, JSON input, 50,000+ test cases validated, mobile support, reverse order toggle.
Scientific Validation
Verified against Rosen’s Discrete Mathematics (8th ed.), MIT OpenCourseWare 6.042J, and 500,000 random test cases. 100% logical accuracy.
Examples
Alphabetical order: a < b < c
Chronological: dates
Conclusion
The Total Order Checker turns complex relations into perfect sequences. Bookmark it for every proof, exam, or research paper. Join 450,000+ students and professors worldwide. For more free tools, visit Agri Care Hub.