relation of medians of two way partition

Question

Given a set of finite real numbers $X \subset \mathbb{R}$, one can obtain the median using $M: X \to \mathbb{R}$.

And $X$ can be partitioned into two subsets $A_i, B_i \subset X, A_i \cup B_i = X , A_i \cap B_i = \emptyset, A_i \neq \emptyset, B_i \neq \emptyset$, without losing generulaity we arrange the subsets such that $M(A_i) \leq M(B_i)$.

For any two different partition $(A_1, B_1)$ and $(A_2, B_2)$, does such two partitions exists such that $M(A_1) \lt M(A_2)$ and $M(B_1) \lt M(B_2)$ ?

Answer 1

• $X = \{ 1, 3, 4, 5, 10\}$
• $A_1 = \{1,5\}$, $M(A_1)=3$
• $B_1=\{3,4,10\}$, $M(B_1)=4$
• $A_2=\{1,3,4,5\}$, $M(A_2)=3.5$
• $B_2=\{10\}$, $M(B_2)=10$

satisfy these inequalities.