$(P(X), C_R)$ may be a choice structure even if $R$ is not a rational relation.

Would you please give me an example to show that $(P(X), C_R)$ may be a choice structure even if $R$ is not rational (i.e., complete and transitive).

Clarification:

  • For any nonempty set $X$, let $P(X)$ denote the set of all nonempty subsets of X.
  • For any nonempty subset $B$ of $P(X)$, a function $c: B \rightarrow P(X)$ is called a choice function iff $c(A) \subset A$ for all $A \in B$. The pair $(B, c)$ is called a choice structure.
  • For any binary relation $R$ on $X$, define the function $C_R : P(X) \rightarrow P(X) \cup \{\emptyset\}$ as follows: $$C_R (A) = \{x \in A : ( \forall y \in A ) ( xRy ) \}.$$

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