Regular and Non-Regular Languages Are all finite Are all infinite languages non-regular? The two most useful ones are closure under: • Intersection
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(a) Union of two non-regular languages cannot be regular Now L1 is regular (since regular languages are closed under complementation) Since, L1 is regular, hence its intersection with L i e L1 ∩ L = L2 is regular (since regular languages are closed under intersection) Therefore, L2 is regular
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Are there two non-regular languages whose concatenation is regular? Show that the intersection of two sets of languages can be empty, finite (of arbitrarily
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Closure Properties of Regular Languages ○ Union ○ Concatenation ○ Kleene star which is non-prime if both factors are greater than 1: (x + z) > 1
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Closure Properties of Regular Languages DFA State Is LREG closed under union? E(0)[i, j]=1 if qi and qj are both accept states, or both non-accept states
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nonempty alphabet So there are many more nonregular languages than there are reg- ular ones EXAMPLE 8 1 The Intersection of Two Infinite Languages
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cardinality, there must be some non-regular languages By the Pigeonhole Principle, two pigeons share a hole, languages is closed under intersection
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Regular and Non-Regular Languages Are all finite Are all infinite languages non-regular? The two most useful ones are closure under: • Intersection
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Once we have defined languages formally, we can consider For example, is the intersection of two regular languages be non-accepting, and make the non-
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regular languages, but are not as easy for other language families 99 construction for the intersection of two regular languages given in Theorem 4 1, finding
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