Question

Define POSET? Draw the Hasse diagram for the partial ordering {(A, B): A ≤ B} on the power set e(S) where S= {a, b, c} and ≤ is subset relation. ​

Answer 1

Answer:

A partially ordered set (or poset) is a set taken together with a partial order on it. Formally, a partially ordered set is defined as an ordered pair ,where is called the ground set of and is the partial order of

Step-by-step explanation:

A Hasse diagram is a graphical representation of the relation of elements of a partially ordered set (poset) with an implied upward orientation. A point is drawn for each element of the partially ordered set (poset) and joined with the line segment according to the following rules:

If p<q in the poset, then the point corresponding to p appears lower in the drawing than the point corresponding to q.

The two points p and q will be joined by line segment if p is related to q.

In above diagram, 3 and 4 are at same level because they are not related to each other and they are smaller than other elements in the set. The next succeeding element for 3 and 4 is 12 i.e, 12 is divisible by both 3 and 4. Then 24 is divisible by 3, 4 and 12. Hence, it is placed above 12. 24 divides both 48 and 72 but 48 does not divide 72. Hence 48 and 72 are not joined.

We can see transitivity in our diagram as the level is increasing.

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