MHB Transitivity as property of relation

[roni1]

In Hebrew, one explain to me that:
"Transitivity is a property (or attribute - I don't which word is correct) of property".
So,
(1) Which word is correct?
(2) Why Transitivity is not standalone by itself?
(3) Are there relations of other kind, that no standalone by themselves?

 

[Evgeny.Makarov]

(1) Which word is correct?

Either "property" or "attribute" can be used. These words are used in their usual sense from the dictionary, not in a technical sense. The second occurrence of "property", though, is probably used in a technical sense.

From the mathematical standpoint, a property, or predicate, on a set $A$ is any subset of $A$. Alternatively, a predicate on $A$ can be defined as a function from $A$ to the set {true, false}. The the subsets consists of those elements of $A$ that are mapped to true.

A binary relation on a set $A$ is any subset of $A\times A$, a Cartesian product of $A$ and $A$, which is a set of all ordered pairs of elements of $A$. Thus, a binary relation is a property of ordered pairs, or a predicate on $A\times A$. Binary relations can be transitive or not. Thus, "transitivity" is a property of a binary relation, i.e., something that can be true or not about this relation.

(2) Why Transitivity is not standalone by itself?

Any adjective, like "transitive", requires a noun in mathematics just like in everyday language.

(3) Are there relations of other kind, that no standalone by themselves?

An adjective "continuous" is a property of functions in calculus, so to talk about continuity you must apply it to some functions.