CSCI 2824 Lecture 19: Properties of Relations
We will now restrict ourselves to relations 
for a set 
. Such relations can be viewed
as a graph.
Graphs and Relations
Relations with domain and co-domain as the same
set can be viewed as a graph.
A graph 
consists of a set of nodes and a relation of edges. Each edge in a graph 
corresponds to a pair 
.
It is a common convention to call the set of nodes 
(rather than
) edge relation 
(rather than 
) if we are talking about a graph. But we will ignore this
convention for now.
We will look at two examples of relations and their corresponding graphs.
Example # 1
Consider the relation 
over the set of nodes 
. Its graph is depicted
below:
Note that the graph has a node 
without any links. This is because
but there is no tuple involving in the relation 
that
describes the edge set.
Next, note that the edges corresponding to 
and 
are
called self-loops. The graph of the relation in this example has two
self loops, one over 
and the other over 
.
Example # 2
Consider the relation 
over the set of nodes 
. Its graph is depicted
below:
Note that the arrow from 1 to 2 corresponds to the tuple 
, whereas the reverse arrow from 
to 
corresponds to the tuple 
.
Types of Relations
We first study three types of relations: reflexive, symmetric and transitive.
A relation is reflexive iff 
for all 
.
From the graph, we note that a relation is reflexive if all nodes in the graph have self-loops
The relation 
from example #2 above is reflexive whereas the
relation 
from example #1 is not. is missing the self loops
from 
and 
.
The next concept is that of a symmetric relation.
A relation is symmetric iff for all , 
.
From its graph, a relation is symmetric if for every “forward” arrow from to 
, there is also a reverse arrow from to .
The relation from example #2 above is symmetric whereas the
relation from example #1 is not. has the edge 
but
not the reverse edge 
.
Finally, we will talk about transitive relations.
A relation is transitive iff for all a,b,c, IF and 
THEN 
.
In graph terms, if we start at some node , and using an edge go from to and then from to 
using another edge, we should also be able to go from to directly using an edge.
The relation from example #2 above is not transitive. It is
missing the edges 
and 
, that would make it transitive.
The relation from example #1 is transitive, on the other hand.
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:
A relation is irreflexive iff 
for all .
For all , we have 
.
While a reflexive relation has all the self-loops, an irreflexive one has no self-loops.
The relation in example # 1 is not irreflexive since it has
self-loops 
d,d)$. Removing these from the relations
yields us an irreflexive relation
Example
Take the set 
. Give us examples of relations that are
Reflexive:
. Irreflexive:
. Caution Irreflexive is not the logical negation of reflexive. It is stronger than that. Symmetric:
. Transitive: example in class.
Equivalence:
. Equivalence-2:
.
What about the empty set as a relation? Is it reflexive? Symmetric? Transitive??
Example
Conider the standard 
relation over 
.
It is reflexive since
for all 
.It is not symmetric:
but 
.It is transitive:
.It is not irreflive since
Anti-Symmetric Relation
We 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 is anti-symmetric iff whenever 
and 
are both in then 
.
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 relation on 
is anti-symmetric. Whenever 
and 
then . In fact, the notion
of anti-symmetry is useful to talk about ordering relations such as 
over sets and over natural numbers.
Partial Orders
A relation is a partial order iff it is
reflexive
transitive
anti-symmetric.
The easiest way to remember a partial order is to think of the relation over sets. In fact, the partial order definition
is an abstraction of the subset relation.
Example-1
Let us take the words in an english dictionary. The lexicographic ordering (or alphabetic ordering) is a partial order.
Example-2
Is the order on a partial order? What about the 
ordering?
Example-3
Is the 
relation on a partial order?
Example-4
Is the empty relation a partial order, in general?
Partial orders can leave elements incomparable. We saw the example with where
is incomparable in the ordering to 
.
A strict partial order is an irreflexive, transitive and anti-symmetric relation.
It is inspired by the 
order between sets which captures the notion of
proper subsets.
Total Order
A 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 relation. It is a partial order over but
additionally for any 
either 
or 
.
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 relation on .
More Examples
Let us take an example and study various find of relations. Let us take the set of natural numbers and classify relations over them.
. 
. 
. 
.
| Relation | Reflexive?? | Symmetric?? | Transitive? | Anti-Symmetric? | Irreflexive? |
| 1 | Y | Y | Y | N | N |
| 2 | N | N | N | Y | N |
| 3 | |||||
| 4 |