Can you tell me how to construct the integer numbers ($\mathbb Z$) as equivalence classes of pairs of natural numbers ($\mathbb N$)? And also tell me the commutative and associative law by an equivalence relation. Be sure to use only addition and multiplication.
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1$\begingroup$ Define $(x,y)\sim (u,v)\,:\Longleftrightarrow\, x+v = u+y$. $\endgroup$– Friedrich PhilippMar 13, 2016 at 4:51
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Yes, this is possible. Consider the relation $(a,b) \sim (c,d) \text{ iff } a+d = c+b$.
Verify that $\sim$ is an equivalence relation and let $[(a,b)]$ be the equivalence class of $(a,b)$ with respect to $\sim$. Define
$[(a,b)] +_\sim [(c,d)] := [(a+c,b+d)]$ and $-_\sim[(a,b)] := [(b,a)]$.
Prove that these are well-defined functions and that $$\pi \colon (\{ [(a,b)] \colon a,b \in \mathbb N \}, +_\sim) \to (\mathbb Z , +), [(a,b)] \mapsto a-b$$ is an group isomorphism.
I leave it to you to define $\cdot_\sim$ such that $\pi$ becomes a ring isomorphism.
answered Mar 13, 2016 at 4:55
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$\begingroup$ So am I solved the problem need to prove that?(Prove that these are well-defined functions and that π:({[(a,b)]:a,b∈N},+ ∼ )→(Z,+),[(a,b)]↦a−b is an group isomorphism. $\endgroup$– hee-suckMar 13, 2016 at 7:17
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$\begingroup$ @hee-suck I'm not entirely sure what you're asking, but verifying all the above lets you conclude that the structure $(\{ [(a,b)] \mid a,b \in \mathbb N \}, +_\sim, \cdot_\sim)$ is a ring that satisfies the properties you would expect $(\mathbb Z, +, \cdot)$ to have. So you may as well take it as the definition for $\mathbb Z$. $\endgroup$ Mar 13, 2016 at 7:43
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