Question 117
Consider any rectangular table having finitely many rows and columns, with a real number $a(r, c)$ in the cell in row $r$ and column $c$. A pair ($R, C$), where $R$ is a set of rows and $C$ a set of columns, is called a saddle pair if the following two conditions are satisfied:
(i) For each row $r'$, there is $r \in R$ such that $a(r, c) \geqslant a(r', c)$ for all $c \in C$;
(ii) For each column $c'$, there is $c \in C$ such that $a(r, c) \leqslant a(r, c')$ for all $r \in R$.
A saddle pair ($R, C$) is called a minimal pair if for each saddle pair ($R', C'$) with $R' \subseteq R$ and $C' \subseteq C$, we have $R' = R$ and $C' = C$.
Prove that any two minimal pairs contain the same number of rows.
(i) For each row $r'$, there is $r \in R$ such that $a(r, c) \geqslant a(r', c)$ for all $c \in C$;
(ii) For each column $c'$, there is $c \in C$ such that $a(r, c) \leqslant a(r, c')$ for all $r \in R$.
A saddle pair ($R, C$) is called a minimal pair if for each saddle pair ($R', C'$) with $R' \subseteq R$ and $C' \subseteq C$, we have $R' = R$ and $C' = C$.
Prove that any two minimal pairs contain the same number of rows.