Answer to Question #141758 in Discrete Mathematics for Asdfg

Question

Let A= {1,2,3,4,6,9,12} let a R b if a is divided by b. Show that R is POSET, Draw Hasse Diagram. Prove or disprove if it is a Lattice.

Answer 1

a divides a so the relation R is reflexive. If $a,b$ are positive integers then, if $a|b$ then clearly, $b\nmid a$ . Hence the relation is not symmetric. Now $a|b\Rightarrow b=ax$ for some integer $x.$ Again $b|c\Rightarrow c=by$ for some integer y. Hence $c=axy$ and so $a|c.$ Hence the relation is transitive. So the relation is a partial order relation and the set is a poset. Also its not totally ordered since 4,9 are non comparable.

Now its not a lattice since $4\vee 9$ doesn't exist. since their least upper bound must be divisible by both 4 and 9 and no such element exist.

The Hasse Diagram is below: