We will define the relation ~ on $\mathbb N \times \mathbb N$ by $(a,b)\sim (c,d)$ iff $a + d = b + c$.

Prove that the operation given by: $[(a,b)][(c,d)] \stackrel{\text{def}}= [(ac + bd, ad + bc)]$ is well-defined.

My attempt at this proof:

Let $(a,b)$ and $(a',b')$ be elements of $[(a,b)]$, and similarly, $(c,d)$ and $(c',d')$ be elements of $[(c,d)]$.

My guess is that we need to show that:

$(a'c' + b'd', a'd' + b'c')$ is an element of $[(ac + bd, ad + bc)]$.

So we know the following:

$$aa' + bb' = ab' + ba' \\ cc' + dd' = cd' + bc'$$

But now I'm wondering what to work with. It would be great if someone could give some assistance. Thank you!

Edit: I am also seeking assistance on the following problems:

Q: Prove that $\mathbb N \times \mathbb N$/~ contains an additive identity, i.e. find an element [(i, j)] ∈ $\mathbb N \times \mathbb N$/~ with the property that

$$[(i,j)] + [(c,d)] = [(c,d)] $$

Q: Prove that every element of $\mathbb N \times \mathbb N$/~ has an additive inverse, i.e. for any $[(a,b)] \in \mathbb N \times \mathbb N/~$, show that there exists $[(c,d)] \in \mathbb N \times \mathbb N/~$ such that

$$[(a,b)] + [(c,d)] = [(i,j)]$$ where [(i,j)] is the additive identity.


What is it all about? We want to introduce the negative numbers, constructed somehow using only the naturals. An integer is represented by $(a,b)$ and it wants to be $a-b$ which not necessarily exist yet (in $\mathbb N$), and now we are only allowed to use $+$ and natural numbers.

How did you get the last 2 equations? They should be $a+b' = b+a'$ and $c+d' = d+c'$..

When $(a,b)\sim(a',b')$, either $a\le a'$ or $a'<a$, because of symmetry, we may also assume the former, if it makes you more comfortable. Then $a'=a+n$ for some $n\in\mathbb N$, and hence, using the defn. of $\sim$, we also have $b'=b+n$. Similarly, we can assume that $c\le c'$ and hence $c'=c+m$ and $d'=d+m$. And.. probably the best is to make one step at one time:

First assume that $C:=(c,d)$ is fixed, and $A:=(a,b)\sim (a',b')=:A'$, then show that $AC\sim A'C$. Finally this same step applies for $A'C\sim A'C'$.

For the rest two questions, guess what integers are represented by the following pairs: $(149,149)$, $(2,7)$, $(7,2)$.. hope it helps

  • $\begingroup$ Makes sense, thank you. Would you be able to assist on the other 2 questions? $\endgroup$ – Julian Park Oct 7 '12 at 16:17
  • $\begingroup$ just post them.. $\endgroup$ – Berci Oct 7 '12 at 16:17
  • $\begingroup$ Dear Berci, they're in this thread. (under the edit). Thank you. $\endgroup$ – Julian Park Oct 7 '12 at 16:21
  • $\begingroup$ ..well.. I would suggest to start them by yourself. Have any ideas? Guess, what could be an $(i,j)$ for the unit of +? What could be the additive inverse of an $(a,b)$? $\endgroup$ – Berci Oct 7 '12 at 16:23
  • 1
    $\begingroup$ Yes. But, what $(i,j)$ pairs represent zero, and, for $(a,b)$, what pair represents "$-(a,b)$"? Note that we still don't have $-$ yet. For example, what integer does $(2,7)$ represent? $\endgroup$ – Berci Oct 7 '12 at 16:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.