Show that the relation $R$ in the set $A = \{x \in Z : 0 \leq x \leq 12\},$ given by $R = \{(a, b) : a = b\}$ is an equivalence relation. Find the set of all elements related to $1$.

(A) The relation is defined as $R = \{(a, b) : a = b\}$ on the set $A = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$.
$1$. Reflexivity: For any element $a \in A,$ we have $(a, a) \in R$ because $a = a$. Thus,$R$ is reflexive.
$2$. Symmetry: Let $(a, b) \in R.$ This implies $a = b,$ which further implies $b = a.$ Therefore,$(b, a) \in R.$ Thus,$R$ is symmetric.
$3$. Transitivity: Let $(a, b) \in R$ and $(b, c) \in R.$ This implies $a = b$ and $b = c.$ Consequently,$a = c,$ which means $(a, c) \in R.$ Thus,$R$ is transitive.
Since $R$ is reflexive,symmetric,and transitive,it is an equivalence relation.
The set of all elements related to $1$ is the set of all $x \in A$ such that $(x, 1) \in R.$ Since $x = 1,$ the only such element is $1.$ Therefore,the set is $\{1\}$.