Reflexive
∀a ∈ A, (a,a) ∈ R
Every element must be related to itself. In a relation matrix, every diagonal entry is 1.
Binary Relation Properties Calculator
Enter the set and relation
Private by design: the set and ordered pairs are processed on this device. They are not uploaded, saved, or added to the page URL.
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Reflexive, symmetric, and transitive definitions
A binary relation on A is any subset R ⊆ A × A. The order of each pair matters: (a,b) and (b,a) are generally different.
Symmetric
(a,b) ∈ R ⇒ (b,a) ∈ R
Every related pair must also occur in the reverse direction. Diagonal pairs satisfy their own reverse.
Transitive
(a,b),(b,c) ∈ R ⇒ (a,c) ∈ R
Every two-step relation chain must have its direct concluding pair.
Irreflexive
∀a ∈ A, (a,a) ∉ R
No element is related to itself. A relation may be neither reflexive nor irreflexive.
Antisymmetric
(a,b),(b,a) ∈ R ⇒ a = b
No two distinct elements may be related in both directions. This is not the opposite of symmetric.
Equivalence relations, partial orders, and closures
Equivalence relation
A relation is an equivalence relation exactly when it is reflexive, symmetric, and transitive. Equivalence relations divide the carrier set into disjoint equivalence classes.
Partial order
A relation is a partial order exactly when it is reflexive, antisymmetric, and transitive. Distinct elements do not need to be comparable.
Reflexive closure
Add every missing diagonal pair (a,a). The number shown by the calculator is the exact number of additions needed.
Symmetric and transitive closures
The symmetric closure adds missing reversed pairs. The transitive closure adds all pairs reachable by a nonempty chain; it is computed to a fixed point.
Worked relation-property example
Let A = {a,b,c} and R = {(a,a),(b,b),(c,c),(a,b),(b,a),(b,c)}.
| Property | Result | Reason |
|---|---|---|
| Reflexive | Yes | All three diagonal pairs are present. |
| Symmetric | No | (b,c) is present but (c,b) is missing. |
| Transitive | No | (a,b) and (b,c) are present but (a,c) is missing. |
Because symmetry and transitivity fail, this relation is not an equivalence relation. It is also not a partial order because the distinct elements a and b occur in both directions, violating antisymmetry.
Input rules, assumptions, and limits
Set elements are case-sensitive text labels. Separate elements of A with commas, semicolons, or line breaks. Put one ordered pair on each relation line. Accepted pair styles are (a,b), [a,b], a,b, a -> b, and a → b. Whitespace around a label is ignored; internal spaces are retained.
Every endpoint in R must already appear in A. Duplicate set elements and duplicate ordered pairs are ignored and reported. Commas, semicolons, arrows, parentheses, square brackets, and line breaks are reserved separators and cannot be part of an element label.
The empty set and empty relation are supported. On the empty set, the empty relation is reflexive, symmetric, transitive, irreflexive, and antisymmetric by vacuous truth. On a nonempty set, the empty relation is not reflexive, but it remains symmetric, transitive, irreflexive, and antisymmetric.
The calculator supports up to 200 set elements and 40,000 nonempty relation lines. It checks every relevant pair and two-step chain, then computes transitive closure with a bounded Boolean matrix. These limits prevent accidental extreme input from freezing the page.
Definitions and terminology follow Mathematics LibreTexts, “Properties of Relations”.
Calculation note: the calculator exhaustively checks the finite pairs and two-step chains represented by the input, including vacuous empty-set cases, and computes transitive closure to a fixed point. Last reviewed: .
Binary relation properties FAQ
How do you know if a relation is reflexive?
A relation R on A is reflexive when every diagonal pair (a,a) belongs to R for every a ∈ A. The calculator lists any missing diagonal pairs.
How do you know if a relation is symmetric?
A relation is symmetric when every pair (a,b) in R has its reverse (b,a) in R. A diagonal pair is already its own reverse.
How do you test whether a relation is transitive?
For every two pairs (a,b) and (b,c) in R, check that (a,c) is also in R. The calculator shows chains whose required concluding pair is missing.
What makes a relation an equivalence relation?
A relation is an equivalence relation exactly when it is reflexive, symmetric, and transitive. All three tests must pass.
What makes a relation a partial order?
A relation is a partial order exactly when it is reflexive, antisymmetric, and transitive. Symmetry is not required.
Is the empty relation symmetric and transitive?
Yes. The empty relation is symmetric and transitive because there are no pairs or chains that violate either implication. It is reflexive only when the underlying set is also empty.
Are symmetric and antisymmetric opposites?
No. Symmetric requires reverse pairs, while antisymmetric forbids distinct elements from occurring in both directions. A relation containing only diagonal pairs can be both symmetric and antisymmetric.
Does this relation calculator store my input?
No. Parsing, analysis, copying, and CSV creation happen locally in your browser. The tool does not transmit or save the set or relation.