Definition:Elementary Equivalence

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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 $\mathcal{N}\models \phi$.