Solution to Let A= {1,2,3,4,6,9,12} let a R b if a is divided by b. Show that … - Sikademy
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Archangel Macsika

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.

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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:

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