Let R be a relation on the plane R2 x ab defined as follow

Let R be a relation on the plane R^2 = {x = (a,b)} defined as follows. Decide if the relation is reflexive, symmetric, transitive, an equivalence relation. If R is an equivalence relation describe the equivalence classes. Let 0 = (0,0).
Below x = (a,b) in R^2 and y = (c,d) in R^2 and xRy if and only if

(a) x,y in a line.

(b) |0x| = |0y| (|0x| = sqrt(a^2+y^2), distance from 0 to x)

(c) |0x|^2 + |0y|^2 = 1.

(d) |xy| = 1. (|xy| is the distance between x and y, i.e. sqrt((a-c)^2 + (b-d)^2) ).

(e) a+b = c+d

(f) ad-bc = 0 (hint: ad-bc is a determinant, if a determinant of a 2 x 2 matrix is zero this means the vectors of the rows are multiples of each other)

Solution

Solved the first two parts, post multiple problems to get the remaining answers

(a) x,y in a line

Relation is reflexive if (a,a) belongs to R for every a in set

(x,x) is not a line since it is the same point, hence it cannot be a line hence the relation is not reflexive

Relative is symmetric if (a,b) belongs to R, then (b,a) must also belong to R

(x,y) is a line is a symmetric relation since if (x,y) represents a line then (y,x) will also represents a line

Relative is transitive if (a,b), (b,c) belongs to R, then (a,c) belongs to R

if (x,y) is line and (y,z) is line, then (x,z) will also be a line

Hence the relation is not reflexive, symmetric and transitive

Hence the relation is not an equivalence relation

b) |0x| = |0y|

relation is reflexive since |0x| = |0x| since distance calculated for the same point will be equal hence the relation is reflexive

Relation is symmetric since |0x| = |0y| => |0y| = |0x| hence the relation is symmetric

Relation is transitive since |0x| = |0y| and |0y| = |0z|, hence it implies |0x| = |0z|

Hence the relation is reflexive, symmetric and transitive

Hence the relation is an equivalence relation

Set 1 = {(0,1), (0,-1), (1,0),(-1,0)} distance 1 class and you can write all other classes with the same procedure