Properties of Relations in their Representations

Topics: Discrete Mathematics

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Let $R$ be a relation and $A$ be its matrix.

We can use $A$ or the relation’s digraph to easily determine the properties of $R$:

1. Reflexive if and only if $A$ has $1$s in its diagonal.
2. Symmetric if and only if all arrows in the digraph are bidirectional.
3. Antisymmetric if and only if between any two vertices there is, at least, one arrow.
4. Transitive if and only if the digraph holds the property that if there exist arrows from $x$ to $y$ and from $y$ to $z$, then there’s one from $x$ to $z$.