Languages' reflexivity and transitivity and Kleene closure

From: ZZambia (mvigliar_at_infinito.it)
Date: 07/10/04 

Date: 9 Jul 2004 16:08:32 -0700

Please help me in solving this formal exercise:

The language X is defined over S* (S alphabet)

Well, X is said to be "transitive" if X^2 is contained in X
Also, X is said to be "reflexive" if E (empty word) is contained in X.

Now, we want do explain why for every language L defined over S*, L*
(the Kleene closure of L) is the smallest transitive and reflexive
language that contains L.
(Namely, by contruction L is included properly in L*)

To do so, we must demonstrate both (1) and (2):

(1) L* is reflexive and transitive.

(2) if L is included in Y, then also L* is included in Y, for every
regular set Y proven to be reflexive and transitive.

Any suggestion welcome in solving (1) and (2) given the previous
facts.

Thanx,
ZZambia

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