CSCI 2824 Lecture 19: Properties of RelationsWe will now restrict ourselves to relations Graphs and RelationsRelations Graph
A graph It is a common convention to call the set of nodes We will look at two examples of relations and their corresponding graphs. Example # 1Consider the relation ![]() Note that the graph has a node Next, note that the edges corresponding to Example # 2Consider the relation ![]() Note that the arrow from 1 to 2 corresponds to the tuple Types of RelationsWe first study three types of relations: reflexive, symmetric and transitive. Reflexive Relation
A relation From the graph, we note that a relation is reflexive if all nodes in the graph have self-loops The relation The next concept is that of a symmetric relation. Symmetric Relation
A relation From its graph, a relation is symmetric if for every “forward” arrow from The relation Finally, we will talk about transitive relations. Transitive Relation
A relation In graph terms, if we start at some node The relation Putting all these together, a relation is an equivalence iff it is reflexive, symmetric and transitive. We now consider the polar opposite of a reflexive relation, an irreflexive relation: Irreflexive Relation
A relation For all While a reflexive relation has all the self-loops, an irreflexive one has no self-loops. The relation ExampleTake the set
What about the empty set as a relation? Is it reflexive? Symmetric? Transitive?? ExampleConider the standard
Anti-Symmetric RelationWe looked at irreflexive relations as the polar opposite of reflexive (and not just the logical negation). Now we consider a similar concept of anti-symmetric relations. This is a special property that is not the negation of symmetric. A relation Anti-symmetric is not the opposite of symmetric. A relation can be both symmetric and anti-symmetric: Another example is the empty set. It is both symmetric and anti-symmetric. The Partial OrdersA relation
The easiest way to remember a partial order is to think of the Example-1Let us take the words in an english dictionary. The lexicographic ordering (or alphabetic ordering) is a partial order. Example-2Is the order Example-3Is the Example-4Is the empty relation a partial order, in general? Partial orders can leave elements incomparable. We saw the example with A strict partial order is an irreflexive, transitive and anti-symmetric relation.
It is inspired by the Total OrderA total ordering is a partial order that also satisfies the property that there are no incomparable elements. The definition of a total ordering is inspired by the A strict total ordering is a strict partial order that ensures that no two different elements are incomparable. Strict total orderings are inspired by the More ExamplesLet us take an example and study various find of relations. Let us take the set of natural numbers and classify relations over them.
|