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Peter Linz Edition 4 Exercise 4.1 Question 19 (Page No. 110)

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Define an operation $third$ on strings and languages as

                          $third(a_1a_2a_3a_4a_5a_6...)=a_3a_6...$

with the appropriate extension of this definition to languages. Prove the closure of the family of
regular languages under this operation.
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Peter Linz Edition 4 Exercise 4.1 Question 26 (Page No. 110)
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Peter Linz Edition 4 Exercise 4.1 Question 21 (Page No. 110)
Define $exchange(a_1a_2a_3...a_{n-1}a_n)=a_na_2a_3...a_{n-1}a_1$, and $exchange(L)=$ {$v:v=exchange(w)$ for some $w∈L$} Show that the family of regular languages is closed under exchange.
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