Take strict preference to be primitive, define weak preference and indifference in terms of strict preference

Question

Question: Take $\prec$ to be primitive and define $\preceq$ and $\sim$ in terms of $\prec$.

Would I write:

$x\preceq y$: $x \prec y$ (or would I not include this here?), not $(y \prec x)$;

$x \sim y$: not $(x \prec y)$ and not $(y \prec x)$?

I know how to take $\preceq$ and define $\prec$ and $\sim$ in terms of $\preceq$ but not starting with $\prec$.

Answer 1

I am not familiar with decision theory, but this looks like it corresponds to constructing $\leq$ and $=$ from $<$. Define $x = y$ := ($x \nless y$ and $y \nless x$), and define $x \leq y$ := ($x < y$ or $x = y$). The trick is to define $=$ first, and to assume that the set we are working over is well-ordered.

Answer 2

Start by assuming that $\prec$ is asymmetric ($x \prec y$ implies $y \not\prec x$) and negatively transitive ($x \not\prec y$ and $y \not\prec z$ imply $x \not\prec z$). Then $\prec$ is a strict preference relation.

Given a strict preference relation $\prec$, you can define the weak preference by $x \preceq y$ if $x \not\prec y$; and you can define indifference by $x \sim y$ is $x \not\prec y$ and $y \not\prec x$.

You need the two properties above to make sure that weak preference and indifference satisfy the usual properties.