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Let 𝑅 be the relation on the set of ordered pairs of positive integers (i.e. ℤ+ × ℤ+) such that ((𝑎, 𝑏), (𝑐, 𝑑)) ∈ 𝑅 is and only if 𝑎 + 𝑑 = 𝑏 + 𝑐. Show that 𝑅 is an equivalence relation. Find the equivalence class of (1,2).

I did show that R is an equivalence relation. Next I tried to find the equivalence class like this: Let [x] be the equivalence class of (1,2). Therefore,

[x] = {(e, f) ∈ ℤ+ × ℤ+ | (e, f)R(1, 2)}

1 + f = 2 + e

From here onwards I don't understand what to do.

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  • $\begingroup$ What do $p$ and $q$ have to do with $e$ and $f$? $\endgroup$
    – aschepler
    Commented Apr 26, 2021 at 17:28
  • $\begingroup$ @aschepler that was a mistake. I corrected it. $\endgroup$
    – S. Tiss
    Commented Apr 26, 2021 at 17:30

1 Answer 1

Reset to default
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$\begin{align}[x]&=[(1,2)]\\&=\{(e,f)\in\Bbb Z^+\times\Bbb Z^+\mid(e,f)\mathrel{R}(1,2)\}&&\text{definition: equivalence class}\\&=\{(e,f)\in\Bbb Z^+\times\Bbb Z^+\mid e+2=f+1\}&&\text{definition: }R\\&=\{(e,f)\in\Bbb Z^+\times\Bbb Z^+\mid f=1+e\}\\&=\{(e,1+e)\mid e\in\Bbb Z^+\}\end{align}$

Please lemme know if that's clear enough.

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  • $\begingroup$ Here is a basic MathJax reference and tutorial. You did make one mistake, which I’ve corrected: $e$ ranges over $\Bbb Z^+$, not $\Bbb R$. $\endgroup$
    – Brian M. Scott
    Commented Apr 26, 2021 at 22:47

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