Total ordering in the context of Transitive relation

In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation ${\displaystyle \leq }$ on some set ${\displaystyle X}$, which satisfies the following for all ${\displaystyle a,b}$ and ${\displaystyle c}$ in ${\displaystyle X}$:

1. ${\displaystyle a\leq a}$ (reflexive).
2. If ${\displaystyle a\leq b}$ and ${\displaystyle b\leq c}$ then ${\displaystyle a\leq c}$ (transitive).
3. If ${\displaystyle a\leq b}$ and ${\displaystyle b\leq a}$ then ${\displaystyle a=b}$ (antisymmetric).
4. ${\displaystyle a\leq b}$ or ${\displaystyle b\leq a}$ (strongly connected, formerly called totality).

Requirements 1. to 3. just make up the definition of a partial order.Reflexivity (1.) already follows from strong connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders.Total orders are sometimes also called simple, connex, or full orders.