Question

Let SS be the Cartesian coordinate plane R×RR×R and define the equivalence relation RR on SS by (a,b)R(c,d)⟺a+2b=c+2d(a,b)R(c,d)⟺a+2b=c+2d.

(a) Find the partition PP determined by RR by describing the pieces in PP.

(b) Describe the piece of the partition that contains the point (5,3)​

Answer 1

We are working in S=R×R=R2S=R×R=R2, which is the Cartesian plane, not R3R3. That said, for every r∈Rr∈R, there exists a line with slope −12−12 such that each point (x,y)(x,y) lies on one and only one line: the line with y-intercept at (0,r/2)(0,r/2).

To see this, note that RR partitions SS into parallel lines of slope −12−12: i.e., the equivalence classes of RR consists of disjoint parallel lines, each of slope −12−12, and so for any given value of r∈Rr∈R, there is a line (a "piece" of the partition of R2R2) consisting of all points (x,y)5(x,y) such that x+2y=rx+2y=r, i.e., all points satisfying the equation of the line:

x+2y=r⟺y=−

So the point (5,3)(5,3) lies on the "piece" of the partition of S=R×RS=R×R that is, in fact, the line

x+2y=11⟺y=−12x+112x+2y=11⟺y=−12x+112

See, for example, a graphic representation of 5 equivalence classes (lines) in the partition of the Cartesian plane S=R×RS=R×R into parallel lines with slope −12−12:

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

Hope it helps

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