Relation with same extension with different properties Suppose there are two relations where  $R' \subseteq A' \times A'$ and $R \subseteq A \times A$ where $A \subset A'$ and $R =  R'$ . For $R$ the condition of reflexivity holds, since $\forall x \in A; xRx$, but $\exists x \in A'; \neg(xR'x)$, so reflexivity does not hold for R'.
Given a principle for equality, $R = R' \Rightarrow \forall \mathfrak{F} (\mathfrak{F}R' \Leftrightarrow \mathfrak{F}R)$. This should hold for the specific instance of reflexivity of a relation.
Are there different properties for extensionally identical objects?
 A: There are two different ways we can think about a relation: as simply a set of ordered pairs, or as a set of ordered pairs together with an explicit domain and codomain. Some properties - like symmetry and transitivity - make sense for the first approach, while others explicitly refer to domain/codomain and so only make sense for the second approach. Reflexivity on the other hand is in this latter category, for exactly the reason you observe. It doesn't make sense to say that a given set of ordered pairs is or is not a reflexive relation.
(This is similar to the situation regarding functions, where we have a choice of whether or not to include an explicit codomain. Some properties of functions, such as surjectivity, only make sense if we include this extra information.)

If we need to keep things straight, we can introduce distinguishing terminology (although this is enough of a non-issue that I don't think there's a standard here). For instance:

*

*A bare binary relation is just a set of ordered pairs.


*A detailed binary relation is triple $(A,B,R)$ where $R\subseteq A\times B$.
Then the point is this:

It makes sense to ask whether a detailed binary relation is reflexive, but it does not make sense to ask whether a bare binary relation is reflexive.

