1. The problem statement, all variables and given/known data 1.Assume the language has equality and a two-place predicate symbol. Given two structures (N;<) and (R;<), find a sentence true in one structure and false in the other. Can these two structures be elementarily equivalent? Can they be isomorphic? Why or why not? 2.The completeness theorem tells us that each sentence either has a deduction or has a counter-model (i.e., a structure in which it is false). For each of the following sentences (in the attached picture), either show there is a deduction or give a counter-model. 2. Relevant equations 1.What is the relation between elementarily equivalent and isomorphic 2. How to decide whether the sentence has a deduction or a counter model? 3. The attempt at a solution In model theory, a field within mathematical logic, two structures M and N of the same signature σ are called elementarily equivalent if they satisfy the same first-order σ-sentences. An isomorphism is a kind of mapping between objects, which shows a relationship between two properties or operations. If there exists an isomorphism between two structures, we call the two structures isomorphic. In a certain sense, isomorphic structures are structurally identical.