Test the follow binary relations on the given sets S for ref

Test the follow binary relations on the given sets S for reflexivity, symmetry, antisymmetry, and transitivity:

S = N x N

(x1,y1)p(x2,y2) <--> x1 ? x2 and y1 ? y2

Solution

If x1 belongs to S , x1<x1 and y1 > y1 is not true , hence S is not reflexive .

x1<x2 and y1 > y2 implies x2<x1 and y2 >y1 is also not true , hence this is not symmetric

x1<x2 and y1 > y2 implies x2<x1 and y2 >y1 . this implies that x1=x1 and y1=y2 . Hence this relation is anti symmetric

If x1<x2 and x2<x3 , x1<x3 and also , if y1 ?y2 and y2>y3 , y1 >y3 is true . Hence this relation is transitive .