Relations

Question

Consider the following relation:

$R={(x,y)| y=x^i, ∃ “i”  in Z }$

R is

i) Reflexive

ii) Symmetric

iii) Anti symmetric

iv) Transitive

Thanks!

Answer 1

A relation if reflexive iff ,

xRx exists . Which is true here since x= x^1 for all x $\in$ Z.Thus it is reflexive.

It is not symmetric , since (2,4) exists but (4,2) does not belong to the Relation .

It is anti-symmetric , because of the reflexive elements.

If xRy exists then y=x^i and if yRz exists then z=y^j . so we can write z = (x^i)^j.

 

Thus the relation is reflexive , anti-symmetric and transitive . Thus a Poset.