Definition:Elementary Equivalence

Definition
Let $\mathcal{M},\mathcal{N}$ be $\mathcal{L}$-structures.

We say that $\mathcal{M}$ and $\mathcal{N}$ are elementarily equivalent if for all $\mathcal{L}$-sentences $\phi$, we have $\mathcal{M}\models \phi$ if and only if $\mathcal{N}\models \phi$.