Let L be a regular language on some alphabet Σ, and let Σ1 ⊂ Σ be a smalleralphabet. Consider L1, the subset of L whose elements are made up only ofsymbols from Σ1, that is,L1 = L ∩ Σ∗1 .Show that L1 is also regular.
Question
Let L be a regular language on some alphabet Σ, and let Σ1 ⊂ Σ be a smalleralphabet. Consider L1, the subset of L whose elements are made up only ofsymbols from Σ1, that is,L1 = L ∩ Σ∗1 .Show that L1 is also regular.
Solution
To prove that L1 is also regular, we can use the closure properties of regular languages. Regular languages are closed under intersection and under homomorphism.
Here are the steps:
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We know that L is a regular language.
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We can construct a regular language L2 = Σ*1. This is a regular language that contains all possible strings over the alphabet Σ1.
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Since regular languages are closed under intersection, the intersection of L and L2, which is L1 = L ∩ Σ*1, is also a regular language.
Therefore, L1 is also a regular language.
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