 ##  [Elementary Equivalence](/elementary-equivalence) 

  ##  [Elementary Equivalence](https://mathlogic.quantumdictionary.io/elementary-equivalence-0) 

  
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- Natural &amp; Formal Sciences -

**Mathematics &amp; Logic Dictionary**

 
Definition

Two structures M and N (same signature) are elementarily equivalent when they satisfy exactly the same first-order sentences (no parameters); symbolically M ≡ N. Equivalence concerns sentences, not necessarily formulas with parameters.

 
 ##  [Elementary Equivalence](https://puremath.quantumdictionary.io/elementary-equivalence-1) 

  
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- Mathematics &amp; Logic -

**Pure Mathematics Dictionary**

 
Definition

A relation between two structures in the same language that they satisfy exactly the same first-order sentences; equivalently, their complete first-order theories coincide.