# 44 - immediate predecessors the other pairs in the partial...

This preview shows page 1. Sign up to view the full content.

5.1 Hasse Diagrams Since partial orders are special binary relations on a set A , we can repre- sent them by directed graphs. However, these graphs get rather cluttered if every arrow is drawn. We therefore introduce Hasse diagrams , which pro- vide a compact way of representing the partial order. First we require some de±nitions. D EFINITION 5.3 If R is a partial order on a set A and a R b for a ± = b , we call a a predecessor of b , and similarly b a successor of a . If a is a predecessor of b and there is no c with a R c and c R b , then a is the immediate predecessor of b . Hasse diagrams are like directed graphs, except that they just record the
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: immediate predecessors; the other pairs in the partial order can be inferred. Also the direction of the lines is usually omitted, with the convention that all lines are directed up the page. We give two examples of Hasse diagrams. E XAMPLE 5.4 For example, the Hasse diagram for the relation ‘is a divisor of’ for the set { 1 , 2 , 3 , 6 , 12 , 18 } is 1 2 3 6 12 18 E XAMPLE 5.5 The Hasse diagram for the binary relation ⊆ on P ( { 1 , 2 , 3 } ) is ∅ { 3 } { 1 , 3 } { 1 , 2 , 3 } { 2 , 3 } { 1 , 2 } { 1 } { 2 } 44...
View Full Document

## This note was uploaded on 03/02/2010 for the course MATH Math2009 taught by Professor Koskesh during the Spring '09 term at SUNY Empire State.

Ask a homework question - tutors are online