$(a,b)\,R\,(c,d)\iff a+2b=c+2d:\;$ Equivalence classes of a Partition

Question

Let $S$ be the Cartesian coordinate plane $\mathbb{R}\times \mathbb{R}$ and define the equivalence relation $R$ on $S$ by $(a,b)\,R\,(c,d)\iff a+2b=c+2d$.

$\hspace{1cm}$(a) Find the partition $P$ determined by $R$ by describing the pieces in $P$.

$\hspace{1cm}$(b) Describe the piece of the partition that contains the point $(5,3)$.

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Could someone explain this to me, please? I suspect that the pieces are just the sets of points in the lines that are the intersections of $z=c$ and $z=x+2y$, but I don't think that is what this question is asking. Could someone explain this visuallyperhaps?

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Answer 1

You’re making it too hard: you need only $\Bbb R^2$, not $\Bbb R^3$. Fix a real number $k$.

• The set of all pairs $\langle x,y\rangle\in\Bbb R^2$ such that $x+2y=k$ forms one equivalence class; why?

• What does $\{\langle x,y\rangle:x+2y=k\}$ look like as a subset of the plane?

• Which of these classes contains $\langle 5,3\rangle$?

Answer 2

We are working in $S = \mathbb R \times \mathbb R = \mathbb R^2$, which is the Cartesian plane, not $\mathbb R^3$. That said, for every $r \in \mathbb R$, there exists a line with slope $-\frac 12$ such that each point $(x, y)$ lies on one and only one line: the line with y-intercept at $(0, r/2)$.

To see this, note that $R$ partitions $S$ into parallel lines of slope $-\frac 12$: i.e., the equivalence classes of $R$ consists of disjoint parallel lines, each of slope $-\frac 12$, and so for any given value of $r \in \mathbb R$, there is a line (a "piece" of the partition of $\mathbb R^2$) consisting of all points $(x, y)$ such that $x + 2y = r$, i.e., all points satisfying the equation of the line: $$x + 2y = r \iff y = \underbrace{-\frac 12}_{\text{slope}}x + \underbrace{\frac 12r}_{y\text{-intercept}}$$

So the point $(5, 3)$ lies on the "piece" of the partition of $S = \mathbb R\times \mathbb R\,$ that is, in fact, the line $$x + 2y = 11 \iff y = -\frac 12 x + \frac{11}{2}$$

See, for example, a graphic representation of 5 equivalence classes (lines) in the partition of the Cartesian plane $S = \mathbb R \times \mathbb R\,$ into parallel lines with slope $-\frac 12$:

The equivalence class in which $(5, 3)$ belongs is the green line plotted above: $x + 2y = 11$.