How should the proof of property (12.3.1) be modified to prove property (12.3.2)?
The property that equivalence is an equivalence relation on , the set of states of for each .
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.
The properties of reflexivity, symmetricity and transitivity of should be proved for any input string in the accepted language of the finite-state automaton as well as the relation. The condition that should be changed for equivalence is the length of input string is less than or equal to where for each .
Suppose are three states of and hereafter the length of input string is less than or equal .
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 input string ,
Hence, and is equal for input string .
Therefore, we can conclude that is symmetric
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