The symmetric difference of two sets S1 and S2 is defined as S1 ⊖ S2 = {x|x ∈ S1 or x ∈ S2, but x is not in both S1 and S2} The nor of two languages is defined as nor (L1, L2) = {w|w ∉ L1 and w ∉ L2}. Which of the following is correct ?
The family of regular languages are closed under both symmetric difference and nor.
Regular languages are closed under union, intersection and complement (via DFA product and complementing the accepting states). Any operation built only from these is therefore also closed.
Step 1 — Symmetric difference: \(L_1 \ominus L_2 = (L_1 \cup L_2) \cap \overline{(L_1 \cap L_2)}\). Union, intersection and complement of regular languages are regular, so the result is regular.
Step 2 — Nor: \(\text{nor}(L_1, L_2) = \overline{L_1} \cap \overline{L_2} = \overline{L_1 \cup L_2}\). The union is regular and its complement is regular, so nor of two regular languages is regular.
Step 3 — A DFA for either operation is the product automaton of the two DFAs; only the choice of accepting pairs differs (exactly one component accepting for symmetric difference, neither accepting for nor).
So any choice saying the family fails to be closed under one or both operations is wrong.
The family of regular languages are closed under both symmetric difference and nor. option is correct.
Similar Questions
Regular expression for the complement of language \(L = \{a^n b^m \mid n \ge 4,\ m \le 3\}\) is
Given two languages :
\(L_1 = \{(ab)^n a^k \mid n \gt k,\ k \ge 0\}\)
\(L_2 = \{a^n b^m \mid n \ne m\}\)
Using pumping lemma for regular language, it can be shown that
Which of the following statements is false ?
The regular expression corresponding to the language L where
\(L = \{ x \in \{0, 1\}^{*} \mid x\) ends with 1 and does not contain substring 00 \(\}\) is :
Let L be any language. Define even (W) as the strings obtained by extracting from W the letters in the even-numbered positions and even(L) = {even (W) | W ∈ L}. We define another language Chop (L) by removing the two leftmost symbols of every string in L given by \(\text{Chop}(L) = \{W \mid \nu W \in L, \text{ with } |\nu| = 2\}\).
If L is regular language then
For the above specified passage, which of the following represents the regular expression?
Which of the following properties correctly describe a Regular Grammar ?
(A) All production rules are of the form A→xB or A→x, where A and B are non terminal symbols and x is a terminal symbol.
(B) Regular grammars are more powerful than context-free grammars and can express any type of language.
(C) There is a direct correspondence between regular grammar and finite automata.
(D) Regular grammars can generate languages that are not recognised by any type of automata.
Choose the correct answer from the options given below :
The precedence of regular expression operators from lowest to highest precedence is:
Which of the following is TRUE about the Pumping Lemma for regular language?
Consider the following two languages :
L1 = {0i1j| gcd (i, j) = 1}
L2 is any subset of 0*.
Which of the following is correct ?
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 $L_1$ 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? Regular expression for the complement of language \(L = \{a^n b^m \mid n \ge 4,\ m \le 3\}\) is
Given two languages :
\(L_1 = \{(ab)^n a^k \mid n \gt k,\ k \ge 0\}\)
\(L_2 = \{a^n b^m \mid n \ne m\}\)
Using pumping lemma for regular language, it can be shown that
- 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)?
