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In this paper, we consider a switching system with several operating modes which represent normal or faulty behavior. The objective is to find a necessary and sufficient condition to distinguish these modes through the input/output data. If faulty operation modes are considered the distinguishability condition corresponds to a fault detectability condition. As in , we define in this paper two notions of distinguishability: the distinguishability through the input/output data of the system and the distinguishability through the parity residuals of each mode. The Necessary and Sufficient Condition (NSC) of distinguishability through parity-space residuals in  was obtained in the case where the parity-space orders are identical in all modes. Here a more generalized NSC of distinguishability is provided for systems which could have different parity-space orders. We show in this context two main results: a NSC of distinguishability through parity residuals which generalizes the NSC in  and the equivalence between the two notions of distinguishability. We also demonstrate that the distinguishability definition in  is linked to the notion of equivalent representations used in the theory of realization.