Given a set $A=\{x \in Z:0 \leq x\leq 12\}$ and relation $R=\{(a,b):a=b)\}$:

Let $a=b$. For every $a \in A (a,a) \in R$, $a=a$. Therefore R is reflexive.

If $(a,b) \in R$ then $a=b$. Similarly, if $(b,a) \in R$, then $b=a$. Therefore, R is symmetric.

If $(a,b) \in R$ then $a=b$ and, if $(b,c) \in R$, then $b=c$.

Now, if $(a,c) \in R$ then $a=c$. Since $a=b$ and $b=c$ $\rightarrow$ $a=c$. Therefore, R is transitive..

Hence $R$ is an equivalence relation as its reflexive, symmetric and transitive.