For similarly, given two Mealy machines, let (Me 1 )(Me 2 ) mean that an input string is processed on Me 1 and then the output string is immediately fed into Me 2 (as input) and reprocessed Only this second resultant output is considered the final output of (Me 1 )(Me 2 ) If the final output string is the same as the original input string, we say that (Me 1 )(Me 2 ) has the identity property, symbolically written (Me 1 )(Me 2 ) identity Given two specific machines such that (Me 1 )(Me 2 ) reproduces the original bit string, we aim to prove (in the following two problems) that (Me 2 )(Me 1 ) must necessarily also have this property Show that the 2 n possible n bit strings when fed into Me 1 give 2 n different outputs

Question: For similarly, given two Mealy machines, let (Me 1 )(Me 2 ) mean that an input string is processed on Me 1 and then the

For similarly, given two Mealy machines, let (Me₁)(Me₂) mean that an input string is processed on Me₁ and then the output string is immediately fed into Me₂ (as input) and reprocessed. Only this second resultant output is considered the final output of (Me₁)(Me₂). If the final output string is the same as the original input string, we say that (Me₁)(Me₂) has the identity property, symbolically written (Me₁)(Me₂) = identity.
Given two specific machines such that (Me₁)(Me₂) reproduces the original bit string, we aim to prove (in the following two problems) that (Me₂)(Me₁) must necessarily also have this property.

Show that the 2ⁿ possible n-bit strings when fed into Me₁ give 2ⁿ different outputs.

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