Is a reflexive relation on the set of real numbers infinite?

Question

In the mathematics textbook which I am studying, it is written that a relation $\mathit{R}$ on a set $\mathit{A}$ is not reflexive if there exists an element $a \in A$ such that $(a, a) \notin R$. So, for the set of real numbers $\mathbb{R}$, is a reflexive relation $\mathit {R}$ defined on it infinite? If it is finite, there must be at least one element $a \in \mathbb{R}$ such that $(a, a) \notin \mathit{R}$, which would again, make it incorrect, am I right?

Answer 1

You can have a transitive and symmetric relation of any cardinality $\leq \mathfrak{c}$. For consider any subset $S \subseteq \mathbb{R}$. Then the relationship $=_S = \{(s, s) \mid s \in S\}$ is transitive, symmetric, and of cardinality $|S|$.