Chapter 2.5, Problem 10E

1. Let $A=Z^2$ and, for $a=\left(a_1,a_2\right)$ and $b=\left(b_1,b_2\right)$ in A, define $a\underset{\_}{\prec }b$ if and only if $a_1\le b_1$ and $a_1+a_2\le b_1+b_2$. Prove that $\underset{\_}{\prec }$ is a partial order A. Is this partial order a total order? Justify your answer with a proof or a counter example.

2. Generalize the result of part (a) by defining a partial order on the set $Z^n$ of n-tuples of integers. (No proof is required).