Fork me on GitHub
Math for the people, by the people.

User login

quote marks should look like ``this''

Primary tabs

quote marks should look like ``this''

An easy way to see that $P_1$ effects a one-to-one correspondence between $\mathbb{Z}_+^2$ and $\mathbb{Z}_+$ is as follows: Define the "successor" of a pair $(x,y) \in \mathbb{Z}_+^2$ to be the pair $(x-1,y+1)$ when $x=0$; otherwise, when $x=0$, the successor is $(y+1,0)$. It is easy to see that every pair has a successor and that every pair except $(0,0)$ is the successor of exactly one other pair. With this definition of successor, the set of pairs of positive integers satisfies the Peano axioms and, hence, is isomorphic to the integers. From the definition of $P_1$ it follows that, if $(x',y')$ is the successor of $(x,y)$, then $P_1 (x',y') = P(x,y) + 1$ and that $P_1(0,0) = 0$. This means that $P_1$ is the isomorphism described two sentences ago.

Parting words from the person who closed the correction: 
Status: Accepted
Reference to the user who closed the correction.: 
Reference to the article this correction is about: 
Status of the article (was it accepted?): 
1
Status of the article (is it closed?): 
1
What kind of correction is this: 
Meta

Subscribe to Comments for "quote marks should look like ``this''"