Question

If A = {1, 2, 3, 4, 6, 12} then define a relation R by aRb if and only if a divides b. Prove that R is a partial order on A.

Answer 1

A binary relation is a partial order if and only if the relation is reflexive, antisymmetric and transitive

$R=\{(1,1),(1,2),(1,3),(1,4),(1,6),(1,12),(2,2),(2,4),(2,6),(2,12),(3,6),$

$(3,12),(4,12),(6,12),(3,3),(4,4),(6,6),(12,12)\}$

R is reflexive: $(1,1),(2,2)(3,3),(4,4),(6,6),(12,12)\isin R$

R is antisymmetric: if $(a,b)\isin R$ then $(b,a)\notin R$ for $a\neq b$

R is transitive: if aRb and bRc then aRc for all $a,b,c\isin A$