# 📝Binary relation properties

Reflexive:

- for every x, x ≤ x
as a graph:

Symmetric:

- For every x and y, if x ≤ y then y ≤ x
- if there is an edge between vertices, there is an edge in the opposite direction
as a graph

Antisymmetric:

- If x ≤ y and y ≤ x, then x = y
- if there is an edge between vertices, there is no other edge in the opposite direction
as a graph:

\begin{tikzcd} a \arrow[loop left] \arrow[r] & b \end{tikzcd}

Asymmetric:

- $\forall a,b \in X : a R b \rightarrow \lnot (b R a)$ (including $a=b$)
- A relation is asymmetric iff it is both antisymmetric and irreflexive
as a graph:

Transitive:

- If x ≤ y and y ≤ z, then x ≤ z
as a graph:

Connexity (connex relation):

- the relation is defined between all pairs of elements
- $\forall x, y \in X, x R y \lor y R x$

Semiconnexity (semiconnex relation):

- the relation is defined between all pairs of
*distinct*elements - $\forall x, y \in X, x \ne y \rightarrow x R y \lor y R x$