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I want to consider the collection of all lines with slope 2, which forms a partition of $\mathbb{R}^{2}$. I need to describe the conditions on which $(x_{1},y_{1}) \sim (x_{1},y_{1}).$

I think the conditions will be that $x_{i} < y_{i}$ for all $i$, and that $2x+1=y$ for all $x$ and $y >0$. Am I on the right track to answering this question? It seems to me like a silly question since the answer is given in the prompt (that all x and y must lie on a line with slope 2, and to me that sounds like a solid condition :) )

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If two points are on the same line with slope 2, and $y$-intercept $b$, then they both satisfy the equation $$ y = 2x + b $$ or, written another way, $$ y - 2x = b$$ So your equivalence relation should be $(x_1,y_1) \sim (x_2,y_2)$ if $$ y_1-2x_1 = y_2-2x_2 $$

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    Thanks, Nick. My only question is - why are we necessitating that points must be on the same line, when the *collection* of lines with slope 2 is what partitions $\mathbb{R}^{2}$? Don't we want to specify conditions that can apply to all lines with slope 2 (and therefore all points, I suppose), instead of a specific subset of the partition (i.e. any particular line)? I hope my question makes sense. – 2017-02-28
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    Usually the notation $a \sim b$ refers to an equivalence relation. An equivalence relation defines the "relationship" between elements which are to grouped together into the same "part" of a partition. – 2017-02-28