My main concern is with the notation around order relations, specifically the use of $\succeq$, I briefly outline relations more generally to set up my thinking. But the questions really starts at the bold 4. Thanks!
- Binary relations on a set $X$ or between two sets $X$ & $Y$ can be denoted by the general form $xRy$.
- These relations will have certain properties such as transitivity, reflexivity etc.
- Using this general notation $xRy$ we would then have to define the condition of the relation, and the set/sets they are onto e.g.
- $x,y \in \mathbb{Z} \ xRy$ if $x+y \in \mathbb{N}$
- In some cases the notation itself defines the relation e.g. $x>y$ for $x,y \in \mathbb{Z}$
4) There are classes of relations that share certain properties. E.g. Order relations, who all share Transitivity.
My question is whether $\succeq$, $\preceq$ function as general notation for order relations, just as $R$ does for Binary relations as a whole, or if they can only be used to define specific order relations.
I have seen this notation by the same lecturer in the following contexts:
- $x \succeq y$ be used to describe a (Weak) preorder. (i.e. transitive and reflexive)
- $x \succeq y$ to discribe preferences in Economics e.g. $x \succeq y$ if $U(x) \ge U(y)$ Where $U$ is the Utlity function s.t. $U:X \to \mathbb{R}$ i.e. this is a (Weak)preorder + Completeness.
- $\succeq$ on $\mathbb{R^n}$, x $ \succeq $ y if $x_i \ge y_i, i = 1..n$ I.e. a Partial order, transitive, reflexive, and antisymmetric - (He noted this was a notation choice, and in a later lecture he noted that in general $\ge$ is used for relations that are antisymmetric i.e. Partial Orders.
Solution 1: $x \succeq y$ is used as a general notation like $xRy$ for when we are referring to order relations specifically.
- If so could I define the Total order relation in the following way?
- $x,y \in \mathbb{R}$, $x \succeq y$ s.t. $x\ge y$ (seems silly and redundant but you get my point).
- And for a (Strict)preorder: $x,y \in \mathbb{R}$, $x\succ y$ s.t. $x \gt y$
Solution 2: $x \succeq y$ is used only for partial orders and (Weak)preorders. This could make sense as technically a partial order, is a (weak)preorder + Antisymmetry in the same way that a Total Order is a partial oder + Completeness. (But by this logic I should be able to use it for total orders also?)
Very much looking forward to feedback, thanks!