Question

Determine whether each of the relations defined below on the set of positive integers is a partial order?

(a) (x,y) ϵ R if x ≥ y

(b) (x,y) ϵ R if 3 divides x+y

Answer 1

(x,y) ER if asy uehere x ytzt. [zt be set of posetive integer order relation Lek us examine R partial there fore R is reflexuve. for all xczt reza hold since sex) ER for all atat. let xe, y Ent ko ya 2 and y a hold ire (24) ER, 8 x) ER then x=y resy and yea this implies so if (x,y) and ox) ER then a=4. hence R is antisymmetrie. let and (Y,Z) belongs Ezt and on), to R. then xxy an d 22 then n>y>, z » x>z this implies & Z) ER. so (26) ER and & Z) ER » & Z) ER therefore R is transitive. therefore R is reflexive, antisymmetric and transitive so order relation, on at R partial b (@y) ER if divides x+y. het atat then 2+ = 2* does not always 3 divedles zx. 3X2. So (2, 2) ER. j he choose X 1 then atx= 1+1=2 therefore for all ex) Er does not hold. re EZT so R is not reflenive. therefore R ů not partial order relation.