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.
Check Relation Properties from Finite universe, then verify Ordered pairs in R. Estimate the Relation Properties Relation properties before computing it again. If Ordered pairs in R changes during Relation Properties, keep Ordered pairs in R fixed. That Relation Properties comparison shows whether Relation properties moves as expected.
Property checks support equivalence classes, orderings, dependency models, state transitions, and proof exercises.
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.
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.