-
Definition:
- $\ge$ is reflexive (自反性) if $x\ge x$ for all $x\in X$.
- $\ge$ is complete (完备性) if for all $x,y\in X$, either $x\ge y$ or $y\ge x$ or both.
- $\ge$ is transitive (传递性) if $x\ge y$, either $y\ge z$ implies $x\ge z$.
-
Q1: complete but not transitive
-
Q2: transitive but not complete
-
Definition:
- is rational (理性) if it is complete and transitive.
- Are these definitions valid? Of course!
- Are these definitions good?
- def for rational
-
Thm 1a: $X$ is finite. If $\ge$ is complete and transitive, then there exists a func $U: X \to \mathbb{R}$ such that $x\ge y$