From Schaum's Outline of Theory and Problems of Discrete Mathematics
Theorem 2.3: Let $R$ be a relation on a set $A$. Then:
$(i)$ $R\cup \Delta A$ is the reflexive closure of $R$.
$(ii)$ $R \cup R^{−1}$ is the symmetric closure of $R$.
I think the theorem is wrong.
For $(i)$, consider set $A = \{1, 2, 3,\}$
$R = \{(1,2) (2,1)\}$
The Reflexive closure of $R$ is $R \cup \{(1,1), (2,2)\}$ and not $R \cup \{(1,1) (2,2) (3,3)\}$
I do agree though, that the Reflexive closure of $R$ reflexive$(R) \subseteq R \cup \Delta A$
For $(ii)$ same argument $R \cup R^{-1}$ is $\Bbb U$ for the example above, $R$ is it's own Symmetric closure. And of course $R \in \Bbb U$.
Am I right? Are the two theorems indeed wrong, or am I missing something.
(i) $R \cup \{(1,1),(2,2)\}$ is not the reflexive closure of $R$. A reflexive relation $S$ on $A$ is a relation such that for all $x \in A$, we have $(x,x) \in S$. However, $(3, 3) \notin R \cup \{(1,1),(2,2)\}$, so $R \cup \{(1,1),(2,2)\}$ is not reflexive.
(ii) You are right in that in your example, $R$ is its own symmetric closure. However, your example also has that $R=R \cup R^{-1}$, so $R \cup R^{-1}$ is the symmetric closure of $R$, even in your example.