Answer to Question #146318 in Discrete Mathematics for Promise Omiponle

Question

Let p and q be distinct positive primes. Define the partial ordering ⪯ of S={1, p, p^2, p^3, p^4, q, q^2, q^3, q^4} as follows: a⪯b if and only if a|b. Draw the Hasse diagram for this partial ordering of S.

Answer 1

In order theory, a Hasse diagram is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set (S, ⪯) one represents each element of S as a vertex in the plane and draws a line segment or vector that goes upward from x to y whenever y covers x (that is, whenever x < y and there is no z such that x < z < y). These lines may cross each other but must not touch any vertices other than their endpoints. Such a diagram, with labeled vertices, uniquely determines its partial order. 

In our case, the Hasse diagram is the following: