Decidable languages are closed under set difference. Recursive Languages Closed Under set difference.

Decidable languages are closed under set difference. Most useful when the operations are sophisticated, yet are guaranteed to preserve interesting properties of the language. Jun 27, 2025 · Statement 3 is true as Turing decidable languages (REC languages) are closed under intersection and complementation. My attempt : False, Since $\text {DCFLs are not closed under union nor intersection}$. The class of decidable languages is closed under complementation. With correct knowledge and ample experience, this question becomes very easy to solve. Shuffle Proof idea: a countable set, so the set of “ ” maps TMs into Σ*, TMs, and hence of Turing recognizable languages is also countable; Turing decidable is a subset of Turing recognizable, so also countable. This means that if a language is decidable, performing these operations on it will result in another decidable language. Further, for a given CFG, there is a direct way to produce a pushdown automaton for the grammar (and thereby the corresponding language), though going the other way (producing a grammar given an automaton) is not as direct. m. Feb 10, 2018 · For example, what is the Boolean closure of context-free languages? • knowning the a closure property of an operation is not satisfied for a given language family, study whether it is decidable that taking the operation on language (s) leads to a language in the family. Sigma^* is RE (TM just checks that the input string either only contains symbols from Sigma or else is empty). ) Question: 1）Prove that the set difference operator is closed under decidable languages. , 3⁄4w " Σ , M either accepts or rejects w) Decidable languages are closed under complement. The recursively enumerable (r. Convince yourself that a language is exaclty then decidable, when both, the language itself and its relative complement are r. Additionally, since the class of DCFLs is closed under complement, the totality problem for DCFLs is decidable. Show that the collection of decidable languages is closed under the following set operations: a. Theorem 1. These are also called Turing-acceptable and Turing-recognizable languages. [1] In theoretical computer science, such always-halting Turing machines are called total Turing machines or algorithms. Unless otherwise noted, the alphabet for all questions below is assumed to be = {0. 2 Draw a TG for L1. 2 Identify each state, qi in T G1 such that there exists a y there is a path from qi to a final state in T G1. [10 points) Show that the set of Turing-decidable languages is closed under the following operations. Shuffle In mathematics, logic and computer science, a formal language is called recursively enumerable (also recognizable, partially decidable, semidecidable, Turing-acceptable or Turing-recognizable) if it is a recursively enumerable subset in the set of all possible words over the alphabet of the language, i. I was thinking I could run R and T in parallel, so I would start running R by processing $a_1$ and then by using non-determinism, I would jump to the NFA T to process $b_1$. A set is closed under an operation if whenever two elements of the set are combined using that operation then the result is also in the set. Regular language Answer: A regular language is defined by a DFA. Therefore the answer is option D. e. Prove/disprove that the class of decidable (resp. either accepts or rejects, but never loops. Regular Languages are closed under an operation op on languages if Jul 9, 2015 · 1) RE is not closed under difference. 6 I'm studying Turing Machines and I've already showed how Turing-Decidable is closed for the operations of Union, Intersection, Concatenation, Complement and Kleene Star. Since regular languages are closed under intersection, union, and complement What are regular languages closed under? Union, intersection, concatenation, complement, star, reverse, prefix, etc. Answers Answer 1 The set difference operator is closed under decidable languages because if languages L1 and L2 are decidable, then their difference L1 - L2 is also decidable, as it can be determined by a Turing machine. b. Proof: Let M1 be a TM which decides L1, and let M2 be a TM which decides L2. In mathematics, logic and computer science, a recursive (or decidable) language is a recursive subset of the Kleene closure of an alphabet. I am aware of following two facts related to two concepts: regular languages and finite sets: Regular languages are not closed under subset and proper subset operations. Kleene Closurec. Jul 23, 2025 · Here in the tables below, D means Decidable, SD means Semi-Decidable and NR means Not-Recursively Enumerable. Shuffle This article is about a class of formal languages as they are studied in mathematics and theoretical computer science. Suppose we assume that R. The interesting part is showing that these machine transformations are possible, but that's not too hard. Jul 23, 2025 · As regular and recursive languages are closed under complementation, option 3 and 4 are decidable problems. Construct C, the product automaton of A and B. So Let L1and L2be {a}. [10 points] Show that the set of Turing-decidable languages is closed under the following operations. Proposition: The recognizable languages are closed under union and intersection. Jan 25, 2025 · The previous chapter identified decidable languages about regular, context-free, and deterministic context-free languages. To show that the regular languages are closed under difference, we only have to note that Question: 5. The language of U Here we show that decidable languages are closed under the five "main" operators: union, intersection, complement, concatenation, and star. Recursive Languages Closed Under concatenation. A language is called decidable if an algorithm can determine whether a given string is a member of that language. These languages are synonymous with recursive languages, meaning that the set of valid strings in these languages can be completely listed by some algorithm. I read a proof on the closure of decidable languages under kleene star. Jun 9, 2015 · Since set difference can be expressed using intersection and complement operators, and since Turing-recognizable languages are not closed under complement, we conclude that Turing-recognizable languages are not closed under set difference. [2] Here we show that decidable languages are not closed under homomorphism. Apr 14, 2015 · T-recog corresponds to semi-decidable (r. Show that the set of Turing-decidable languages is closed under the following operations. Every regular language is decidable. Recursive Languages Closed Under intersection with a regular language. e if L is decidable then so is bar (L)) (6) (2pts) Show that the set of Turing-recognizable languages is closed under: Intersection Sep 20 2025 03:18 AM. Kl Mar 6, 2015 · Showing that Turing-recognizable languages are closed under union Ask Question Asked 10 years, 6 months ago Modified 8 years, 2 months ago } A language is decidable if and only if it is Turing-recognizable and co-Turing-recognizable Assume language A is decidable. Recursive Languages But we know that RHS is not closed under set difference so LHS is also not closed under intersection. Context free languages are not closed under complementation, option 2 is undecidable. A: Introduction: In formal language theory, a set difference operation is denoted by the symbol "−" and… Q: Create a Turing machine that computes a mapping reduction from L1 ≤ m L2 where L1 ＝ {w : w ∈ {a,b}*… A: The complete answer is given below . We would like to show you a description here but the site won’t allow us. Terms in this set (4) what is turing decidable languages closed under Union Intersection Concatenation Star Complement Give an informal definition of the term turing-decidable. Convince yourself that there are r. Prove the contrapositive: the complement of a decidable language is decidable. (B) If L, is undecidable and L, is decidable then the symmetric difference of L1 and L2 is undecidable. E is closed under intersection then, Since L1 ∩ L2 = comp (comp (L1) ∪ comp (L2)), this would imply the CFL's are closed under intersection. *Hint: for star, try dynamic programming. We’ll soon see examples of languages that are in RE but not in Dec. Set Differenceb. [13] the regular operations: K ∪ L, concatenation ⁠ K ∘ L {\displaystyle K\circ L} ⁠, and Kleene star L Key Takeaways Regular languages are closed under union, concatenation, Kleene star, intersection, complementation, difference, and reverse operations. For computer languages that allow a function to call itself recursively, see Recursion (computer science). (Since TM halts on all input, we can have a different TM that flip-flops reject/accept. ACFG is decidable. The link below is a good resource to read some more on the topic. languages that are not decidable (e. Closure Under Difference If L and M are regular languages, then so is L – M = strings in L but not M. If you remember dealing with problems asking you to decide if a language is regular, the logic for decidability is quite similar. Regular Languages are closed under an operation op on languages if Decidable languages refer to a class of problems or decision problems for which there exists a Turing machine that halts on any input, providing a yes or no answer effectively. Shuffle May 12, 2023 · The set difference operator is closed under** decidable languages **because if languages L1 and L2 are decidable, then their difference L1 - L2 is also decidable, as it can be determined by a Turing machine. This means that if we start with decidable languages and apply these operations, the resulting languages will also be decidable. A Turing Machine decides a language if it rejects every string it doesn’t accept – i. Logical systems are decidable if membership in their set of logically valid formulas (or theorems) can be effectively determined. complementation. These are also called theTuring-decidable or decidable languages. Given TMs M1, M2 that decide languages L1, and L2 A TM that decides L1 [ L2: on input x, run M1 and M2 on x, and accept i (Similarly for intersection. Examples of decidable languages include regular languages and context-free languages. Understanding the closure properties of CFLs helps in determining which operations preserve the context-free nature of a language. We still have to see whether or not there are recognizable languages that are not decidable, and whether or not there are languages that are not recognizable. Properties of Decidable Languages Theorem (Closure Properties of Decidable Languages) The class of decidable languages is closed under Union Intersection Complementation Concatenation Star Answers Answer 1 The set difference operator is closed under decidable languages because if languages L1 and L2 are decidable, then their difference L1 - L2 is also decidable, as it can be determined by a Turing machine. Yes, the intended meaning there is "If the set L is a decidable language, than the complement of that set is also a decidable language under the algorithm 1. , the universe of regular languages is closed under these operations De nition 1. Properties of Decidable Languages Theorem (Closure Properties of Decidable Languages) The class of decidable languages is closed under Union Intersection Complementation Concatenation Star Mar 27, 2024 · Explanation: Recursive languages are closed under intersections, unions, complement, concatenation, Kleen star and set difference. Apr 6, 2017 · Decidable and Turing-recognizable languages closed under subtraction [closed] Ask Question Asked 8 years, 5 months ago Modified 8 years, 5 months ago Show that the class of Turing-decidable languages is closed under symmetric difference, where the symmetric difference of A and B is the set of elements belonging to either A or B but not both. Halteproblem) Assume that the class of r. Let C be a TM which makes a copy of the input: (s, # w #) Recap: Recognizable versus Decidable Languages A language L is called Turing-Recognizable if there exists a TM M such that L(M) = L 1⁄4 Note: M need not halt on all inputs but it should halt and accept all and only those strings that are in L; it can reject strings by either going to q rej or by looping forever Users with CSE logins are strongly encouraged to use CSENetID only. This brings me to the definition of a Turing Recognizable Language : Def : A Language is called Turing Recognizable if some Turing Machine recognizes it. The set is the set of all decidable languages. Ask Question Asked 7 years, 6 months ago Modified 7 years, 6 months ago Looping and Defining Languages The possibility of looping affects how we define the relationship between a machine and the language it defines TM M recognizes language L iff the strings L put M into the Accept state the strings NOT IN L EITHER put M into the Reject state OR cause M to loop the strings L put M into the Accept state the strings second, show the set of all languages is uncountable e. Lets start with some definitions:- Decidable language -A decision problem P is said to be decidable (i. Decidable Problems In computer science and mathematics, a language is a set of strings made up of symbols from an alphabet. A set S is closed under an operation f Answer: S is closed under f if applying f to members of S always returns a member of S. Mar 4, 2014 · Decidable languages are closed under a few operations, such as set union, set intersection, set complementation, string concatenation, and Kleene closure. Undecidable can be either Semi-Decidable or Not-Recursively Enumerable language. for a given string s, decide if it is in L (trivial by assumption) 2. Suppose that I have 2 DFAs and have 7 and 6 states respectively, and 3 and 4 final states respectively. 4 Non-closure properties SS and S is a new → 1} Context-free languages are not closed under intersection or complement. and 3. Comment. Show that decidable languages are closed under the following operations: a. Next I did some demonstrations to show how T-Recognizable languages are closed for Union, Intersection, Concatenation and Kleene Star. CFLs are closed under some operations, meaning 1 Closure properties of semi-decidable languages Recall that the class of regular languages is closed under union, intersection, complemen-tation, concatenation and so on, but CFLs are only closed under some of these operations (union and concatenation, but not intersection or complementation). Decidable languages, on the other hand, are languages for which there exists an algorithm that can determine whether a given input string is in the language or not. 1). We also show that the same languages ARE closed under inverse homomorphism. Jul. , if A were language of all strings beginning with 0 over alphabet {0,1}, its characteristic sequence would be Complementation : This is fairly straightforward, but the point to note is that turing recognizable languages are NOT closed under com-plementation, while turing decidable languages are. Kleene’s theorem The set of all context-free languages is identical to the set of languages accepted by pushdown automata, which makes these languages amenable to parsing. We will see how the recursive languages differ from recursively enumerable, what are the properties of computable, non-computable, countable, and uncountable sets. True. difference Suppose L1 – L2 is a context free if L1 and L2 are context free. This chapter will study undecidable problems and techniques for proving undecidability. [1] We have described constructions which show that applying these operations to decidable languages result in decidable languages. False, that should be recursive enumerable but not recursive. In this case, we say that the Language is recognized by the Turing Machine. d. and 2 There are 2 such states, q1 a Thm. S be a decidable language. g. But then any RE language would be recursive ( decidable ). star. For example, because the decidable languages are closed under union and complementation, we immediately have that they are closed under intersection and symmetric difference. A homomorphism is a function h from a set A to Jun 30, 2016 · Of course, regular languages aren't closed under every closure property. Proposition Decidable languages are closed under union, intersection, and complementation. languages is closed under complementation and derive a contradiction to the facts mentioned in 2. Sep 10, 2025 · Set difference selects all of the elements in one set that are not present in a second set. e. However, I also know that Turing-recognizable languages are not closed under complement, and $\overline {\bar {A} \cup \bar {B}} = A \cap B = C$, which seems to suggest that Turing-recognizable languages are not closed under intersection. Flipping the accept and reject states generates a TM to decide the complement of this language. For example, If two integers are added, subtracted or multiplied then the sum, difference or product is also an integer. The regular languages are closed under various operations, that is, if the languages K and L are regular, so is the result of the following operations: the set-theoretic Boolean operations: union K ∪ L, intersection K ∩ L, and complement L, hence also relative complement K − L. 2 I have written a proof to show that a Turing Decidable languages are closed under union (amongst other things). Context-free languages are closed under the various operations, that is, if the languages K and L are context-free, so is the result of the following operations: 1. intersection. L1 L2 = {a}, and so is regular. Recursive Languages Closed Undercomplementation. Jul 5, 2023 · We have shown that decidable languages are closed under set difference, Kleene closure, and shuffle. Jul 10, 2024 · Recursive languages VS recursively enumerable languages in tabular form Recursive Languages Closed Under union. What would be the case for semi-definite languages? Here we will show that they are closed under Feb 7, 2025 · Are decidable languages closed under any operations? Yes, decidable languages are closed under operations such as union, intersection, concatenation, and complementation. Run on w. Decidable and Semi-Decidable Languages What is a word and what is a language: a brief reminder. c. Chapter 4: Properties of Regular Languages We begin by looking at closure properties. The key is to ass The regular languages are closed under all usual operations (union, intersection, complement, concatenation, star). union. Decidable languages are closed under union, intersection, and complementation. Prove that the set of polynomial-time decidable languages ( P ) is closed under union, complement, concatenation, and star. This will be shown later. 14 Show that the collection of decidable languages is closed under the following operations. L - M = L ∩ C(M) reduces the closure under set-theoretical difference operator to closure under complementation and intersection. The decidable languages are closed under complement and intersection, so they are closed under difference. The closure and decision properties make regular languages a robust tool for formal language processing and Complementation : This is fairly straightforward, but the point to note is that turing recognizable languages are NOT closed under com-plementation, while turing decidable languages are. 1. Prove that the class of Turing-recognizable languages is closed under Concatenation Given: Two Turing-recognizable languages A and B, and TM’s that recognize them, MA and MB. Study with Quizlet and memorize flashcards containing terms like Regular Languages are closed under:, Context Free Languages are closed under:, Decidable Languages are closed under: and more. , if there exists a Turing machine which will enumerate all valid strings of the language Summary Theorem: A language L is decidable if and only if it is both recognizable and co-recognizable: D = RE ∩ coRE Question: Show that decidable languages are closed under: - set difference - Kleene closure (hint: use recursion) 1 Closure Properties 1. There is clearly a contradiction somewhere in my reasoning. Kleene Closure c. ) either accepts. The statement you quote is an argument why the proof for showing that $\mathrm {R}$ (the set of recursive languages) is closed against complement does not work for $\mathrm {RE}$ (the set of recursively enumerable languages). Jan 25, 2025 · Since DPDAs are PDAs, decidable problems for context-free languages are decidable for DCFLs. Mar 10, 2025 · Decidable Undecidable 1. Dec 21, 2014 · If I have two languages that aren't Turing-recognizable, is the union between them always not T-recognizable? Why? Deterministic context-free languages can be recognized by a deterministic Turing machine in polynomial time and O (log 2 n) space; as a corollary, DCFL is a subset of the complexity class SC. Proof: Let A and B be DFA’s whose languages are L and M, respectively. I'm just not sure if the same applies to intersection of the complement, and how to extend this to any two different languages unless i'm missing something sorry? GeeksforGeeks | A computer science portal for geeks Jul 12, 2025 · Note: There are few more properties like symmetric difference operator, prefix operator, substitution which are closed under closure properties of regular language. ). The recursive languages = the set of all languages that are decided by some Turing Machine = all languages described by a non-looping TM. Submitted by Carmen B. However, there are undecidable Mar 13, 2021 · The interlaced machine show that Turing recognisability is closed under union. Another example is string reversal: if a language A is decidable, then AR is also decidable, because a Most useful when the operations are sophisticated, yet are guaranteed to preserve interesting properties of the language. So L union R cannot be decidable for undecidable L and finite R. Recursive Languages Closed Under set difference. Equivalently, a formal language is recursive if there exists a Turing machine that decides the formal language. , it never loops The recursive languages = the set of all languages that are decided by some Turing M Machine hi = all looping TM. partially decidable) languages is closed under symmetric difference. 4. Jul 23, 2025 · Context-Free Languages (CFLs) are an essential class of languages in the field of automata theory and formal languages. ) The general principle that the above theorem highlights is that since decidable languages are closed under complementation, if L L is an undecidable language, then L = Σ ∗ ∖ L L = Σ∗ ∖L must also be undecidable. Closure properties for decidable languages Theorem The decidable languages are closed under union. May 31, 2017 · Can we use the proof of turing-decidable languages being closed under complement to show that turing-decidable languages are closed under set difference? U uses a fixed, finite set of states, and set of alphabet symbols, but still simulates TMs with arbitrarily many states and symbols. Today: A variety of operations which preserve regularity { i. The There are a variety of other operations under which the decidable languages are closed. (A) The set of decidable languages is closed under symmetric difference. Theorem: Turing decidable languages are closed under intersection. In Computer Science: a word is any combination of symbols, and a language is any set of words. Decidable languages are closed under complement. , have an algorithm) if the language L of all yes instances to P is decidable Mar 8, 2011 · A Language of a Turing Machine is simply the set of all strings that are accepted by the Turing Machine. b) Is it possible that L1 L2 is regular? Yes. We can therefore construct a grammar for (L ∩ ̄L′) ∪ ( ̄L ∩ L′) and use the algorithm for the emptiness problem P∅. Recursive Languages Closed Under intersection. Understanding decidable languages is fundamental in Aug 3, 2023 · Identifying languages (or problems*) as decidable, undecidable or partially decidable is a very common question in GATE. It begins by saying that the turing machine we want to find would non-determistically split the input string and then use the original decider of the language to approve the partition of each branch. The set of all recursively enumerable languages is: closed under complementation closed under of all recursive languages an uncountable set Show that the class of regular languages is closed under shuffle. Your UW NetID may not give you expected permissions. recursive languages, what are the decidable and closure properties of formal languages. Where is it? Oct 3, 2017 · Since decidable languages are closed under set difference and union, this means L must be decidable, a contradiction. (a) union (b) complementation (c) intersection (d) concatenation (e) star [Hint: Consider constructions similar to what we used for Yes. That means the set of decidable languages is closed under these operations. , L M A), and halts on every input (i. Let Sigma be the alphabet of an arbitrary RE language L. A symmetric difference of sets A and B is the set (A \ B) ∪ (B \ A). 3. ∈ iff is decidable and 4 days ago · 5 + Users Viewed 0 + Downloaded Solutions Virginia, US Mostly Asked From (5) (2pts) Show that the set of decidable languages is closed under: Complementation (i. Are there any "natural" closure properties that don't apply to the regular languages? The regular languages are closed under intersection, union and complement, and we know algorithms for these operations. Decidable languages refer to a class of problems or decision problems for which there exists a Turing machine that halts on any input, providing a yes or no answer effectively. Proof: Assume it was. 1 Decidable Languages Boolean Operators Proposition 1. All encoded using the fixed symbols, decoded during emulation. Statement 4 is true as Turing recognizable languages (RE languages) are closed under union and intersection. Oct 9, 2013 · Why are decidable languages closed under complement? So if L is decidable why is the complement of L also decidable. ll llanguages d described ib d b by a non- These are also called theTuring-decidable or decidable languages. On the other hand, the class of semi-decidable languages is not closed under complementation. concatenation. （Do not copy existing ans on chegg） The universal Turing machine U accepts an TM encoding M, w of a TM M and string w, then simulates the execution of M on w. If a language is a decidable there is a TM that accepts and halts strings that belong to the language and rejects and halts strings that do not belong to the language. A theory (set of sentences closed under logical consequence) in a fixed logical system is decidable if there is an effective method for determining whether arbitrary formulas are included in the theory. Can you explain for option $ (1)$, is DCFL are closed under Intersection with Regular Languages? Somewhere, it explained as $\text {DCFL are closed under Intersection with Regular Languages}$. However, when one integer is divided by another Study with Quizlet and memorize flashcards containing terms like What are regular languages closed under?, What are context free languages closed under?, What are deterministic context free languages closed under? and more. Recall that a language is decidable if there exits a decider (Turing machine) that for any input w either accepts or rejects it (never loops). Nov 7, 2015 · I know that the decidable are close under: complementation, union, intersection and concatenation? What about the undecidable languages? I think they are close under complementation, but not under Feb 26, 2018 · Show that of Turing decidable languages is closed under concatenation. Question: Problem 2 (Closure property of P ). All usual decision problems (word problem, emptiness, finiteness, intersection, equivalence) are decidable for regular languages. Question: Show that decidable languages are closed under the following operations:a. If RE would be closed under difference, in particular Sigma^* - L would be RE. ) languages = the set of all languages that are the language of some Turing Machine. Mar 20, 2025 · In addition, we shall arrive to relation between decision problems vs. if possible give examples of languages that are and that aren't turing decidable. [3] The set of deterministic context-free languages is closed under the following operations: [4] complement inverse homomorphism right quotient with a regular language pre: pre ( L {\displaystyle L} ) is Aug 21, 2021 · According to my understanding: Turing-recognizable languages are languages whice are accepted by a Turing machine; decidable languages are languages for which a Turing machines halts, i. Summary A language L is decidable if and only if it is both recognizable and co-recognizable: D = RE ∩ coRE RE Decidable Context-free Here we show that regular languages are closed under complement, in that if L is a regular language, then L' (the set of all strings not in L) is also regular. You do not need to actually construct TMs to accept Question: Show that decidable languages are closed under the following operations: a. Decision properties of regular languages include emptiness, finiteness, membership, equivalence, containment, and intersection emptiness. The direction in which we read a string should be of supreme irrelevance, so for recognizable languages to form a reasonable class they should be closed under reversal. But by the previous result, the set of all languages is uncountable. This would make me think that decidable languages include Turing-recognizable languages, and not viceversa. 11, 2022 07:42 p. Set Difference b. Show that the collection of decidable languages is closed under the operation of a. One example is "closure under addition of all strings of the form $0^n1^n$," but the fact that this brings us outside the regular languages is trivial and uninteresting. Complement: S = { x ∈ Ω | x 6∈S} = Ω − S, where Ω is the universe of all elements under consideration. Clearly, any decidable language is recognizable. Proof. Although it might take a staggeringly long time, M will eventually accept or reject w. It also halts on all input, and accepts the complement of A. Complement. Feb 7, 2025 · Examples of undecidable languages include the Halting Problem and the Post Correspondence Problem. if s is in L, then s is not in its complement, and vice versa". (a) union (b) complementation (c) intersection (d) concatenation (e) star Hint: Consider constructions similar to what we used for regular languages. Unless otherwise noted, the alphabet for all questions below is assumed to be = {0,1}. Regular languages are also closed under symmetric diference and the symmetric diference of two languages is empty if an only if they are equal. 1 Closure Properties 1. Engineering Computer Science Computer Science questions and answers Unless otherwise noted, the alphabet for all questions below is assumed to be £ = {0,1}. Q: Give a constructive proof that set difference is a closure for regular languages. Later, I have written a proof to show that Turing Recognizable languages are closed under union. It is decidable whet Since decidable languages are closed under complement, union, and intersection it directly follows that decidable languages are closed under symmetric difference. Approach: If A and B are regular, then there exists an NFA R and T that recognizes them. Regular. 6) Let L1 and L2 be any two undecidable languages. Feb 25, 2019 · Also that decidable languages are closed under intersection and complement. CS 301 Lecture 18 – Decidable languages recognizes A (i. We will use Dec to name this set. Intersection. They are generated by context-free grammars (CFGs) and are recognized by pushdown automata (PDAs). 4: If L1 and L2 are regular languages, then L1=L2 is regu-lar: The family of regular languages is closed under right quotient with a regular language. Understanding decidable languages is fundamental in Solution for w that decidable languages are closed under the following operations: a. phdp rynmqnz hmjt egqb svsg fsrrvrw ajr zckjnj urwhx rqug