Question

Let A={2,3,4,6,10,12,16} and R is a partial order relation defined on A where aRb if and only if a evenly divides b. 1) Draw the Hasse diagram for R. 2) The minimal elements of R are. 3) The maximal elements of R are.

Answer 1

The Hasse diagram for set A with division as a partial order would depict the comparative divisibility of elements. The minimal elements of A are 2 and 3, and the maximal elements are 16 and 12.

Step-by-step explanation:

The question involves drawing a Hasse diagram for a partial order relation, identifying minimal and maximal elements within a set A. Set A comprises the elements {2,3,4,6,10,12,16}, and the relation R is such that aRb if and only if a divides b evenly. To address the question:

1. Hasse Diagram: The Hasse diagram would represent the elements of A as nodes and the divisions as edges, without implying transitivity (e.g., if 2 divides 4 and 4 divides 12, an edge from 2 to 12 is not drawn because the relationship from 2 to 4 and then from 4 to 12 implies it).
2. Minimal Elements: These are elements of A that are not divided evenly by any other element in A. In this case, the elements 2 and 3 are minimal because there are no other elements in the set which divide them.
3. Maximal Elements: Maximal elements are those that do not divide any other element in the set evenly. For the given set A, 16 and 12 are maximal elements, because there are no elements in A that they divide into without leaving a remainder.