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This question is partly about the difference between orders, preference relations and binary relations in the context where they are similar, but mainly about the use of the associated operator.

The operator $\geq$ can mean greater than or equal on domains where there is a defined sense of cardinality. However, it can also be use to define a (partially) order set $(S, \leq)$, where it means one item is ranked below another. Similarly we have the operator $\succeq$ to express (weak) preference of one element over another(used in economics e.g.). Moreover, the name in latex of this operator (succeq) suggest the notion of precedence, although I couldn't find an example that is explicitely concerned with precedence/succedence. Lastly we have the case of sets with a binary relation $R$ where $aRb$ means $a$ is ranked not lower than be, with the extension $aPb$ meaning that $aPb$ ranks strictly higher than b.

Im confused by the fact that we use these operators one equivalent sets such that the operators are defined for all pairs of elements, I do not see the difference between these operators and there interpretations.

To give an example: $3\geq 2$ when $2,3 \in \mathbb{R}$ is a common expression. Also we can say $a \geq b$ in a partially order set. However i've never seen $2P3$, or $2 \succeq 3$ used while it seems to me they mean the same. So why use different operators that essentially express the same thing in different situations?

I'm guessing it has something to do with domains and/or implied axioms but i can't find out which exactly.

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