I have a question whether the below choice correspondence satisfies WARP or not. I know how to check a choice satisfies WARP when set X is in he form of X={x,y,z} however i cannot figure out when it is in the form given below. Can you help me?
By the way question is coming from Rubinstein's Lecture Notes in Microeconomic Theory book pg.44 Q3.)
C(A) = {x ∈ A | the number of y ∈ X for which V (x) ≥ V (y) is at least |X|/2}, and if the set is empty, then C(A) = A.
Is C(A) satisfy WARP?
HINT: Let $X=\{a,b,c,d\}$, with utility function $V:X\to\Bbb R$ given by $V(a)=B(b)=1$ and $V(c)=V(d)=2$. Find distinct $3$-element subsets $A$ and $B$ of $X$ and distinct $x,y\in A\cap B$ such that $x\in C(A)$, $y\in C(B)$, and $x\notin C(B)$. Note that since $X$ has $4$ elements, for any $x\in A\subseteq X$ we have $x\in C(A)$ if and only if there are at least $2$ elements $z\in A$ such that $V(x)\ge V(z)$.