Show that if the sentence “for all languagesL, L_, ifLandL∪L_ are regular, thenL_ is regular” were true, then all languages would be regular.
Are the following true? Prove or give a counter example.
(a) Each subset of a regular language is regular.
(b) Each regular language has a regular proper subset.
(c) IfLis regular, then so is {xy:x∈L, y_∈L}.
(d) IfRis any set of regular languages, then ∪Ris also regular.
(e) The language {wuR:u,w∈Σ∗} is regular.
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