Problem 5. Are the following statements true or false? Either prove the statemen

Question

Problem 5. Are the following statements true or false? Either prove the statement or give a counter example. (a) f L U L? is regular then L and L are regular. (b) If L is regular then L is regular. (c) If LL? is regular then L and L? are regular. (d) If L and L? agree on all but a finite number of strings, then one is regular if f the other is regular. (e) If R is regular, L is not regular. and L and R are disjoint, then L U R is not regular. (f) If L differs from a non-regular language A by a finite number of strings F. then L itself is not regular.

Explanation / Answer

a.) If L U L' is regular then L and L' are regular

   It is true. Let L= any regular and L'=complement(L),which is also regular.Then L U L' is regular.

b.) If L* is regular then L is regular

   It is true because if L* is regular then L is regular because the subset of L* should be regular and L is part of that subset.

c.) If LL' is regular then L and L' are regular

   it is true.

d.) If L and L' agree on all but a finite number of strings,then one is regular iff the other is regular

   False.Such strings are not in L.

e.) If R is regular,L is not regular, and L and R are disjoint,then L U R is not regular

   False.Closure properties of regular class.

f.) If L differs from a non-regular language A by a finite number of strings F,then L itself is not regular

   False.A regular language has a finite number of prefix equilance class.

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