Definable set

In mathematical logic, a definable set is an n-ary relation on the domain of a structure whose elements are precisely those elements satisfying some formula in the language of that structure. A set can be defined with or without parameters, which are elements of the domain that can be referenced in the formula defining the relation.

Let ${\displaystyle {\mathcal {L}}}$ be a first-order language, ${\displaystyle {\mathcal {M}}}$ an ${\displaystyle {\mathcal {L}}}$-structure with domain ${\displaystyle M}$, ${\displaystyle X}$ a fixed subset of ${\displaystyle M}$, and ${\displaystyle m}$ a natural number. Then:

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