What it shows
A relation R on a set A is any collection of ordered pairs (a, b) from A × A; we write a R b when (a, b) is in R. This page lets you build one on A = {1, 2, …, n} (n from 2 to 6) by tapping a table or typing pairs, and draws it as an arrow diagram: an arrow from a to b for each pair, and a small loop when a R a.
It then checks the four properties you meet in school, and for every “No” it shows the exact pair that breaks the rule:
- Reflexive: every a has a R a (every number has a loop).
- Symmetric: whenever a R b, also b R a (arrows come in both directions).
- Transitive: whenever a R b and b R c, also a R c (every two-step path has a shortcut).
- Antisymmetric: a R b and b R a only when a = b.
A relation that is reflexive, symmetric and transitive is an equivalence relation, and the page lists its equivalence classes. Reflexive, antisymmetric and transitive makes a partial order. It also says whether R is a function from A to A, and whether that function is one-one and onto.
It follows relations and functions in Class 11 maths and the types of relations and functions in Class 12. Sets come first in the venn diagram calculator calculator; the logic simplifier works with the same “and”, “or” and “if … then” that the definitions use, and the combinatorics calculator counts the arrangements behind the numbers of relations and functions.
How to use it
- Choose the size of A and a starting relation, such as a ≤ b or a − b is even.
- Tap squares to add or remove pairs (row a, column b), or type pairs like (1, 2), (2, 3) and press Enter.
- Read the checks. A red “No” comes with the pair that breaks the rule.
- Press reflexive, symmetric, transitive or an equivalence to add the fewest pairs that give R that property (its closure). Added pairs show in orange.
Worked example
On A = {1, 2, 3, 4}, let a R b when a − b is even. Then R = {(1, 1), (1, 3), (2, 2), (2, 4), (3, 1), (3, 3), (4, 2), (4, 4)}.
- Reflexive: a − a = 0 is even, so yes.
- Symmetric: if a − b is even, so is b − a = −(a − b). Yes.
- Transitive: (a − b) + (b − c) = a − c, and the sum of two even numbers is even. Yes.
So R is an equivalence relation, with classes {1, 3} (the odd numbers) and {2, 4} (the even numbers). It is not antisymmetric, because both (1, 3) and (3, 1) are in R.
Now start from b = a + 1, which gives {(1, 2), (2, 3), (3, 4)}. It fails transitivity: (1, 2) and (2, 3) are in R but (1, 3) is not. Press transitive and the page adds (1, 3), (1, 4) and (2, 4), which turns it into a < b.
Counting relations
A × A has n² pairs, and each is in R or not, so a set of n elements has 2^(n²) relations. Of these, 2^(n² − n) are reflexive and 2^(n(n + 1)/2) are symmetric. The number of equivalence relations is the Bell number (1, 2, 5, 15, 52, 203 for n = 1 to 6), and there are nⁿ functions from A to A.
Check yourself
Five questions with fresh numbers: spot a property, complete a transitive relation, count equivalence classes, count relations and count one-one functions. Nothing is saved.
FAQ
Can a relation be symmetric and antisymmetric? Yes: a = b (only loops) is both. The empty relation is too.
Is the empty relation transitive? Yes. There is no a R b and b R c to break the rule.
Is this from my textbook? It covers the standard school topic with its own examples. It is an independent study aid.
Where the numbers come from
- Standard definitions of relations on a set: reflexive, symmetric, antisymmetric and transitive relations, equivalence relations and classes, partial orders, closures (Warshall's method for the transitive closure) and functions. Written for fibub; no textbook text, figures or exercises are used.
- Checked by counting every relation on 3 and 4 elements (512 and 65,536 relations) and comparing with a Python brute force: 171 and 3994 transitive, 5 and 15 equivalence, 19 and 219 partial orders.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.