Question 5
Given $n$ are real numbers $x_{1}, x_{2}, \ldots, x_{n}$ and $y_{1}, y_{2}, \ldots, y_{n}$ respectively. The elements of a $n \times n$ matrix $A$ are defined as follows: ($1 \leq i, j \leq n$)
$$
a_{ij}= \begin{cases}1 & \text{if} x_{i}+y_{j} \geq 0 \\ 0 & \text{if} x_{i}+y_{j}\lt 0\end{cases}
$$
Furthermore, let $B$ be a $n \times n$ matrix with elements $0$ or $1$, such that the sum of the elements in each row and each column of $B$ is equal to the sum of the elements in the corresponding row or column of $A$.
Prove that $A=B$ then holds.
$$
a_{ij}= \begin{cases}1 & \text{if} x_{i}+y_{j} \geq 0 \\ 0 & \text{if} x_{i}+y_{j}\lt 0\end{cases}
$$
Furthermore, let $B$ be a $n \times n$ matrix with elements $0$ or $1$, such that the sum of the elements in each row and each column of $B$ is equal to the sum of the elements in the corresponding row or column of $A$.
Prove that $A=B$ then holds.