Would you please help me to prove the following proposition:
If $(P(X), c_R)$ is a choice structure that satisfies WARP, then $R_c$ is a rational binary relation?
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: $$C_R (A) = \{x \in A : ( \forall y \in A ) ( xRy ) \}.$$
- Weak Axiom of Revealed Preference (WARP): $c(A) \cap B \neq \emptyset \Rightarrow c(B) \cap A \subset c(A)$.