Prove property (12.3.1).

To prove:

The property that R* is an equivalence relation on S, the set of states of A.

Given information:

A is a finite-state automaton and the set of the states of A is denoted by S.

Proof:

R* is the notation that is used to denote ∗− equivalence of two states.

Suppose s,t and u are three states of S.

Suppose the states s and t are ∗− equivalence states of A. Then these two states send the automaton to a nonaccepting state or an accepting state for any input string w in the set of strings. This property can be denoted by sR*t.

If sR*t for any string input w, the eventual function N* will be,

N*(s,w)⇔N*(t,w)

If N*(s,w) is a nonaccepting state, then N*(t,w) is also a nonaccepting state.

By the symmetricity of the above relationship, for any input string w ,

N*(t,w)⇔N*(s,w)

Hence, sR*t and tR*s is equal for any input string w.

Therefore, we can conclude that R* is symmetric.

Also, suppose sR*t and tR*u for any input w in the set of input strings for the automaton A