Consider the set $M = \{1, 2, 3\}$ and the relation $R = \{(1, 1), (2, 2)\}$. Does this relation satisfy reflexivity with respect to the concept of an "equivalence relation"?
Someone told me that for an relation to satisfy reflexivity, it is not necessary for every element $a ∈ M$ to have $(a, a) ∈ R$, as long as $a$ doesn't appear in any pair in the relation $R$. (i.e. in the above example, since $3$ doesn't appear in any pairs like $(3, 1)$, $(1, 3)$, $(3, 2)$, ...etc. then $(3, 3)$ isn't necessary to belong to $R$) Thus, R above satisfies reflexivity.
However, I don't think that's correct because, as I remember, in order for an relation to satisfy reflexivity, then for every $a ∈ M$, there must be $(a, a) ∈ R$
To expand on the points made in the comments: the reflexive property of a relation does require a “domain” from which the elements come. This is in contrast to the other two properties of an equivalence relation (symmetry and transitivity).
In your example, the relation $R = \{(1,1),(2,2)\}$ is a reflexive relation on $\{1,2\}$, but is not a reflexive relation on $\{1,2,3\}$.
Perhaps of additional note is that there is a notion of a quasi-reflexive relation on a set, defined by the rule that $(x,y) \in R \implies (x,x),(y,y) \in R$. It is slightly more general in that we only require elements to be related to themselves, if they are related to elements other than themselves. Your example relation would be quasi-reflexive on $\{1,2,3\}$ as a result.