Let R be a relation on the plane R^2 = {x = (a,b)} defined as follows. Decide if

ID: 3420654 • Letter: L

Question

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)

(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)

Explanation / Answer

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

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