Prove property (12.3.5).
Each k -equivalence class is a subset of a - equivalence class for every integer .
The given statement is, “Each k -equivalence class is a subset of a - equivalence class for every integer ”.
Let k be a positive integer. Let A be an automaton and let be a k -equivalence class of A.
Let s and t be two states in .
Since s and t are two states in the same k -equivalence class, s and t are k -equivalent.
We know that if two states are k -equivalent, then they are also - equivalent. Thus, s and t are - equivalent.
This then implies that there exist a - equivalence class that contains both s and t
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