Prove property (12.3.1).
The property that is an equivalence relation on , the set of states of .
is a finite-state automaton and the set of the states of is denoted by .
is the notation that is used to denote equivalence of two states.
Suppose are three states of .
Suppose the states and are equivalence states of . Then these two states send the automaton to a nonaccepting state or an accepting state for any input string in the set of strings. This property can be denoted by .
If for any string input , the eventual function will be,
If is a nonaccepting state, then is also a nonaccepting state.
By the symmetricity of the above relationship, for any input string ,
Hence, and is equal for any input string .
Therefore, we can conclude that is symmetric.
Also, suppose and for any input in the set of input strings for the automaton
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