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)
(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)
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
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.