Binary Relation

• $R: \{\{a,b\} : a, b \in A\} \to [0, 1]$
• Compares elements in a set $A$ (e.g. $aRb$ where $a,b\in A$)
• $R$ is anything that take in two parameters (Examples: $<, >, \le, \ge, \dots$ )
• Visualizing: In $aRb$, treat $a, b$ as vertices, and $aRb$ as a directed edge, draw a graph.
• Example

Binary Relation Properties

Irreflexive $\Leftrightarrow (\forall a \in A, a\cancel{R}a)$ (no self loops)

• Reflexive $\Leftrightarrow (\forall a \in A, aRa)$ (all self loops) Note: Reflexive is not the negation of irreflexive.

Asymmetric $\Leftrightarrow (\forall a, b \in A: aRb \Rightarrow b\cancel{R}a)$ (all edges single-directional)

• Symmetric $\Leftrightarrow (\forall a,b \in A: aRb \implies bRa)$ (all edges bi-directional)
• Antisymmetric $\Leftrightarrow \forall a,b \in A: (aRb \land bRa \implies a = b)$ (e.g. $\neq$) Contrapositive: $\forall a,b \in A: (a \ne b \implies a\cancel{R}b \lor b \cancel{R}a)$
• Asymmetric ⇒ Antisymmetric
• Asymmetric ⇒ irreflexive

Transitive $\Leftrightarrow (\forall a,b,c: aRb \land bRc \Rightarrow aRc)$

• $aRb,bRa,a\cancel{R}a$ is not transitive
• Irreflexive + Transitive ⇒ Asymmetric

Connected $\Leftrightarrow (\forall a,b \in A: a\ne b \Rightarrow aRb \lor bRa)$

• Doesn’t mean that graph is connected

Partial Order: Reflexive + Antisymmetric + Transitive (e.g. $\le$, divides)