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After showing that the relation $R$ defined on $\mathbb R\times \mathbb R$ by $((a,b), (c,d))\in R$ if $|a|+|b|=|c|+|d|$ is an equivalence relation (I already did), how can I describe geometrically the elements of the equivalence classes $[(1,2)]$, respectively $[(3,0)]$?

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HINT: A pair $\langle x,y\rangle$ belongs to $[\langle 3,0\rangle]$ if and only if $|x|+|y|=|3|+|0|=3$; what’s the graph of the equation $|x|+|y|=3$? More generally, for each $c\ge 0$ you have an equivalence class consisting of all $\langle x,y\rangle$ such that $|x|+|y|=c$; what does the graph of that equation look like?

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The equivalence class is the set of points in $(x,y)\in \mathbb{R}^2$ such that $|x|+|y|=3.$ Try picking some various branches of the $| \cdot |$ and plot those lines to see what it looks like.

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Do you know any important properties of equivalence classes. One is, as it pertains to this example: $$\bigcup_{[(x,y)]\in R}^{} [(x,y)] = \mathbb{R} \times \mathbb{R}$$ This should give tell you that $[(1,2)] = [(3,0)]$ for starters. To get an idea behind the geometry, try starting from the point $(1,2)$ and begin moving parallel to one of the axes and think about how you need to adjust the other coordinate to find other members of $[(1,2)]$.

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