Definition:Definable

Definable Element
Let $\mathcal{M}$ be an $\mathcal{L}$-structure with universe $M$.

Let $A$ be a subset of $M$.

and let $\bar b$ be a $n$-tuple of elements from $M$.

Let $\mathcal{L}_A$ be the language formed by adding constant symbols to $\mathcal{L}$ for each element of $A$.

$\bar b$ is definable over $A$ if there is an $\mathcal{L}_A$-formula $\phi(\bar x)$ with $n$ free variables such that the set $\{\bar m \in M^n : \mathcal{M}\models \phi(\bar m)\}$ contains $\bar b$ but nothing else.

Alternative Definition
The following definition is sometimes used. It is not equivalent in general. However, these definitions are equivalent when working in a saturated model. This is proved in Definable iff Singleton Orbit.

$\bar b$ is definable over $A$ if every $A$-automorphism is an $A,b$-automorphism.

Definable Set
We say that the subset $A$ is a definable set in $\mathcal{M}$ if there is a formula $\phi(x)$ such that $a \in A$ iff $\mathcal{M}\models\phi(a)$.