Question 19 Pre-Write

19. a. Using our definition of the integers $\mathbb{Z}$ as a quotient of $\mathbb{N}\times\mathbb{N}$, propose an appropriate definition of $<_{\mathbb{Z}}$.

$(m, n)<_{\mathbb{Z}}(p, q){\leftrightarrow}m+q<_{\mathbb{N}}n+p$

b. Prove that your proposal is well defined and extends the linear order $<_{\mathbb{N}}$ on the natural numbers.

The way we define integers is through ${\mathbb{N}}$. If $a, b {\in}{\mathbb{N}}, then (a, 0), (b,0){\in}{\mathbb{Z}}$. It follows that if $a<b {\in}{\mathbb{N}}, then (a, 0)<(b,0){\in}{\mathbb{Z}}$.

To show that the proposal is well-defined, we want to show that if $(a, b)<(c, d)$ then $(a{\sim}, b{\sim})<(c{\sim}, d{\sim})$.

page revision: 4, last edited: 15 Mar 2018 01:44