All the relations
|A|=n |A×A|=n²
A={1,2,3} A×A={(1,1),(2,2),(3,3) , (1,2),(2,1),(1,3),(3,1),(2,3),(3,2)} n + n²-n
| Reflexive | Irreflexive | Symmetric | Antisymmetric | Asymmetric | Transitive | |
|---|---|---|---|---|---|---|
| Cardinality of smallest relation | n | 0 | 0 | 0 | 0 | 0 |
| Cardinality of largest relation | $n^2$ | $n^2-n$ | $n^2$ | $n+\frac{n^2-n}{2}=\frac{n(n+1)}{2}$ | $\frac{n^2-n}{2}$ | $n^2$ |
| Number of relations | $2^{n^2-n}$ | $2^{n^2-n}$ | $2^n*2^{\frac{n^2-n}{2}}$ | $2^n*3^{\frac{n^2-n}{2}}$ | $3^{\frac{n^2-n}{2}}$ | No formula |