Question 8
Let $m,n$ be positive integers. Each cell $(r,c)$ of an $m\times n$ table contains a real number $a(r,c)$. Choose a row set $R\subseteq\{1,2,\ldots,m\}$ and a column set $C\subseteq\{1,2,\ldots,n\}$. Call $(R,C)$ a good pair if both conditions hold:
- Condition 1. For every $r'\in\{1,2,\ldots,m\}$, there exists $r\in R$ such that $a(r,c)\ge a(r',c)$ for every $c\in C$.
- Condition 2. For every $c'\in\{1,2,\ldots,n\}$, there exists $c\in C$ such that $a(r,c)\le a(r,c')$ for every $r\in R$.
A good pair $(R,C)$ is minimal if every good pair $(R',C')$ with $R'\subseteq R$ and $C'\subseteq C$ satisfies $R'=R$ and $C'=C$. Prove that the row sets of any two minimal pairs have the same number of elements.
- Condition 1. For every $r'\in\{1,2,\ldots,m\}$, there exists $r\in R$ such that $a(r,c)\ge a(r',c)$ for every $c\in C$.
- Condition 2. For every $c'\in\{1,2,\ldots,n\}$, there exists $c\in C$ such that $a(r,c)\le a(r,c')$ for every $r\in R$.
A good pair $(R,C)$ is minimal if every good pair $(R',C')$ with $R'\subseteq R$ and $C'\subseteq C$ satisfies $R'=R$ and $C'=C$. Prove that the row sets of any two minimal pairs have the same number of elements.