Suppose there are four sets $A, B, C, D$ with each having median $a < b < c < d$.
And the median of any two union. $a< median (A\cup B) < b $.
But I'm wondering would a similar behavior hold for more than 2 set ?
e.g. would $median (A \cup B \cup C) < median (B \cup C \cup D)$ this hold ?
Assuming finite sets. For infinite sets, we need a probability distribution.
Let the non-empty sets be $A_i$ with median $a_i$. In each set let the set of points $x \leq a_i$ be denoted by $L_i$ and the other points be denoted by $U_i$.
Let $X$ be any non-empty set with median $x$. WLOG $x \geq a_i$.
Claim $a_i \leq$ median($X \cup A_i$) $\leq x$.
$\frac{|A_i|}{2}$ points are less than $a_i \leq x$, and $\frac{|X|}{2}$ points are less than $ x$, which implies that $\frac{|A_i| + |X|}{2}$ points are less than $ x$, which implies median($X \cup A_i$) $\leq x$. Similarly show median($X \cup A_i$) $\geq a_i$.
