Question

Show that if l is regular, so is lr.
a) True
b) False

Answer 1

The reverse of a regular language L, denoted as L^R, is also regular. This is because regular languages are closed under reversal, and a finite automaton can be constructed for L^R by reversing the transitions of the automaton that recognizes L.

Step-by-step explanation:

If L is a regular language, its reverse, LR is also regular. This is a fundamental property of regular languages, based on the closure properties of regular sets. Regular languages are closed under several operations, including union, concatenation, and Kleene star, as well as reversal.

To show that the reverse of a regular language is regular, we can use the fact that for any regular language L, there exists a finite automaton that recognizes it. By reversing the directions of the transitions in this automaton and switching the initial and final states (and making the necessary adjustments for multiple initial states), we can create a new finite automaton that recognizes LR. This process effectively demonstrates that LR is recognizable by a finite automaton, and therefore it is regular.

Answer 2

To demonstrate that the reverse of a regular language L (denoted L^r) is also regular, one can construct a nondeterministic finite automaton (NFA) that recognizes L^r by reversing the transitions of the deterministic finite automaton (DFA) for L. Since NFAs can be converted to DFAs, which are equivalent in power, L^r is regular.

Step-by-step explanation:

We need to show that if L is a regular language, then the reverse of L, denoted as Lr, is also regular. A regular language is a language that can be expressed with a regular expression and can be recognized by some finite automaton.

For any regular language L, there exists a deterministic finite automaton (DFA) that recognizes it. To show Lr is regular, we can construct a nondeterministic finite automaton (NFA) that recognizes Lr by reversing all transitions of the DFA for L, making the initial state of the DFA the single final state of the NFA, and making all final states of the DFA into initial states of the NFA. Since NFAs and DFAs are equivalent in their expressive power, and since a NFA can be converted into a DFA, this new DFA will recognize Lr. Hence, Lr is also regular.