Lesson 3

A binary relation \(\preceq\) on a set \(X\) will be called a partial order relation if the following three properties are satisfied:

• reflexivity:  \(x \preceq x\) for all \(x \in X\);


• antisymmetry:  if \(x \preceq y\) and \(y \preceq x\), then \(x=y\) for all \(x,y \in X\);


• transitivity:  if \(x \preceq y\) and \(y \preceq z\), then \(x \preceq z\) for all \(x,y,z \in X\).

The pair \((X,\preceq)\) will be called a partially ordered set (or just poset). We will say that \(x,y \in X\) are comparable if \(x \preceq y\) or \(y \preceq x\). Note that the definition does not require any two elements to be comparable.