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Elementary reactions are the step by step sequences that make up overall reaction mechanisms. For example 3 elementary steps can add up to one overall reaction mechanism with intermediates, catalysts, etc.. The slow step of a mechanism determines the rate law and the overall reaction formula can be determined from the reaction mechanism.
To answer the second part of your question: the decomposition of O3 to O2 can be hypothesized as occurring by different reaction mechanisms that are each made up of different elementary reactions. The decomposition can occur by a one-step reaction mechanism that is composed of just one elementary reaction between O3 molecules to produce O2, or it can takes place by a two-step mechanism where O acts as an intermediate between the two elementary reactions which make up this mechanism. Hope that helps!
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