Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R Assertion A: If L is regular, then its compliment L' is necessarily regular. Reason R: Complement of a language can be obtained by swapping final and non-final states in a DFA. In the light of the above statements, choose the most appropriate answer from the options given below
Analyzing Assertion A: Regular Language Complementation
Assertion A states that if a language $L$ is regular, its complement $L'$ must also be regular. This is a fundamental closure property of regular languages. Regular languages are closed under complementation. This means that for any regular language $L$, its complement $L'$ (containing all strings over the alphabet $\Sigma$ that are *not* in $L$) is also a regular language.
Therefore, Assertion A is correct.
Analyzing Reason R: DFA Complement Construction
Reason R explains a method to obtain the complement of a language using a Deterministic Finite Automaton (DFA). If we have a DFA $M = (Q, \Sigma, \delta, q_0, F)$ that accepts a regular language $L$, we can construct a new DFA $M' = (Q, \Sigma, \delta, q_0, Q \setminus F)$ that accepts the complement language $L'$.
- $Q$ is the set of states.
- $\Sigma$ is the input alphabet.
- $\delta$ is the transition function.
- $q_0$ is the start state.
- $F$ is the set of final (accepting) states for $L$.
- $Q \setminus F$ is the set of non-final states (i.e., all states that are not in $F$).
By swapping the roles of final and non-final states, the new DFA $M'$ accepts exactly those strings that cause the original DFA $M$ to enter a non-final state at the end of the input. This process effectively accepts the complement language $L'$.
Therefore, Reason R is correct.
Evaluating the Relationship between A and R
Reason R provides a concrete construction method (swapping final and non-final states in a DFA) that demonstrates why the complement of a regular language is indeed regular. This construction is the basis for proving the closure property mentioned in Assertion A.
Since both A and R are correct statements, and R directly explains the mechanism behind A, Reason R is the correct explanation for Assertion A.
Conclusion
Both Assertion A and Reason R are correct, and Reason R provides the correct explanation for Assertion A.
Important Questions from Regular Languages
Which of the following are correct on regular expressions?
A. φ + L = L + φ = L
B. εL = Lε = L
C. φL = Lφ = φ
D. φL = Lφ = L
Choose the correct answer from the options given below:
- Let $\Sigma = \{a, b, c\}$. For $x \in \Sigma^*$, and $a \in \Sigma$, let $\#_a(x)$ denote the number of occurrences of $a$ in $x$.
Which one or more of the following option(s) define(s) regular language(s)? - Let $L_1$ and $L_2$ be two languages over a finite alphabet, such that $L_1 \cap L_2$ and $L_2$ are regular languages.
Which of the following statements is/are always true? The Kleene Star operation accepts the following string of finite length over set A = {0,1} | where string s contains even number of 0 and 1.
