Test reflexivity, symmetry, antisymmetry, and transitivity on a finite universe. A checkable formula accompanies relation properties instead of leaving an unexplained number.
The sample is reflexive and symmetric but fails transitivity because some required composed pairs are absent.
A binary relation is a subset of U×U; its ordered pairs determine several independent structural properties.
Treat Finite universe and Ordered pairs in R as parts of the Relation Properties definition rather than interchangeable entries, especially when transferring data from another source.
Save extra digits internally if relation properties will become an input to another calculation.
Property checks support equivalence classes, orderings, dependency models, state transitions, and proof exercises.
Preserve the Relation Properties formula beside Finite universe and Ordered pairs in R; that small record makes a later independent calculation possible.
Adding one pair can repair one property while breaking another. This is a stronger check than judging Relation Properties only by the length or appearance of its output.
Match the reported detail to Finite universe, Ordered pairs in R, not to the amount of text the browser can display. Preserve the exact finite structure whenever it communicates Relation Properties more clearly than a summary label.
Specialized pages combine the required properties for equivalence relations and partial orders. The formula panel makes the chosen definition explicit.
The minimum audit trail is short: Finite universe, Ordered pairs in R, their stated roles, and the defining rule beside the answer. It is enough to distinguish this calculation from a similar-looking shortcut.
Use the universe to test diagonal pairs, reversed pairs, forbidden mutual pairs, and every two-step chain. Relation Properties also connects to order test.
Record a missing diagonal, missing reverse, forbidden mutual pair, or broken two-step chain for every failed property.
The four properties answer different questions. Reflexivity inspects only diagonal pairs; symmetry inspects reversals; antisymmetry restricts mutual pairs between distinct elements; transitivity inspects composable chains. Keeping those witness types separate makes a failed property easier to repair and avoids treating symmetry and antisymmetry as simple opposites.
Symmetric and antisymmetric are not opposites; a relation can satisfy both when only diagonal mutual pairs occur. If the Relation Properties assumptions do not fit, consider equivalence test.
A binary relation is a subset of U×U; its ordered pairs determine several independent structural properties.
Property checks support equivalence classes, orderings, dependency models, state transitions, and proof exercises.
Symmetric and antisymmetric are not opposites; a relation can satisfy both when only diagonal mutual pairs occur.