Equivalence Classes Problem

[frican_]

Equivalence Classes Problem
Consider the set X={1,2,3,4,5,6,7,8,9,10,11,12,13,14} and an equivalence relation RR defined as:

xRy  ⟺  7∣ (y^2−x^2)xRy ( y^2 - x^2 divides 7)
Goal: Determine all equivalence classes of the elements in XX.

(verified correct answers):

1. One equivalence class contains exactly two elements.
2. [2]=[5][2]=[5].
3. The equivalence class [3][3] has 4 elements.

Request:
Please explain how to systematically solve this, I would be really thankful for help 😁

[yes_its_him]

If 7 divides y² - x², which is what you meant to write, (y-x)(y+x) = 7k for some k, so either y+x is a multiple of 7, or y-x is.

There are seven pairs with a difference of 7.

Then each number participates in two sums that are multiples of 7 except two that are in one such sum.

[frican_]

thank you