In this section we restrict ourselves to talking about relations on a set
*A* and define some properties a relation may have.

A relation on *A*
is said to be *reflexive* if for each *a* is related to *a*.
If we let R denote the relation then we have *a*R*a* for each .
An example of a non reflexive relation is the relation "is the father of"
on a set of people. As no person is the father of themself the relation is
not reflexive. As another example consider the relation on
defined by if is odd. Then 1 1
and 3 3 but 0 0 and so the relation is not reflexive.

A relation on *A*
is said to be *irreflexive* if for each *a* is not
related to *a*. This is not the negation of the definition of reflexive.
The relation "is the father of " is irreflexive.

A relation R on *A* is *symmetric* if given then
. The relation "is the sister of" is not symmetric on a set that
contains a brother and sister but would be symmetric on a set of females.
The empty relation on a set is an example of a symmetric relation since
the statement "if *a*R*b*" is always false.

A relation R on *A* is *antisymmetric* if given then
. Again, it is possible to have relations that are neither
symmetric nor antisymmetric.

A relation R on *A* is *transitive* if given *a*R*b* and
*b*R*c* then *a*R*c*. The relation "is an ancestor of" on a set of people
is transitive as is the empty relation on a set.

For a finite set, if we use a table to represent a relation then it is easy to spot some of these properties provided we list the column elements across the top of the tabe in the same order as the row elements down the table. The relation is reflexive if the diagonal entries are all 1 and irreflexive if all the diagonal entries are zero. The relation is symmetric if the lower triangle below the diagonal is the reflection across the diagonal of the upper triangle. It is antisymmetric if whenever the (a,b) position is 1 then the (b,a) position is 0. Note that it is OK to have both positions 0 in this case. Unfortunately, we can not observe transitivity so readily.

Wed Nov 20 00:52:35 PST 1996