Definition of Relation of a Set The standard definition of a relation of an arbitrary set A is a subset of the set product of A, AxA.  Is it okay to define relation R to be a subset of the set product AxA such that R has at least one property P (i.e. inequality, equality, difference, etc.), where any two element of a relation obeys P, and any two element of a set A either obeys P or not?  The standard definition of a relation seems to me little ambiguous.

Well I personally disagree with your sense of ambiguity that you feel.  I think this definition perfectly encapsulates the intuitive notion of a relation.  For example, for $a, b \in A$, the statement "$a$ is related to $b$" can be written in shorthand as $(a,b)$.  A "relation" is then a specific way of grouping these related pairs together, so it seems intuitive to me to make $R \subseteq A \times A$ containing these $(a, b)$'s.  Your addition specification of a property doesn't seem necessary to me.  
