Suppose the language, L, over an alphabet Σ0 is regular, and f : Σ0 → Σ1, is a function from the alphabet Σ0 to the alphabet Σ1. The function f can be lifted to words producing the function: ˆf : Σ∗0 → Σ∗1 ˆf(ε) = ε ˆf(aw) = f(a)ˆf(w) Show that the language Lf = {ˆf(w) ∈ Σ∗1| w ∈ Σ∗0} is regular. Hint: You have to use the formal definition of DFAs.

Q: Prove that the following language A is decidable (e.g., give a TM that decides A). A = {⟨M⟩ | M is…

A: We will prove that language A = {⟨M⟩ | M is an NFA and L(M) = B}, where B is a regular language, is…

Q: Let L = {w#w | w ∈ {a, b}*} be a language. Use the pumping lemma for regular languages to prove that…

A: L = {w#w| w∈ {a, b}*} string generated by this languageare aaab#aaab, babb#babb proof: If possible…

Q: Define a CFG that generates the language over the alphabet Σ = {α,β,θ} defined by the regular…

A: The given regular expression is : β(α + β)*αθα(α + β + θ)*β We need to find context-free grammar for…

Q: Consider a set of symbols = {abc}. (1) Find all strings in E* shorter than 3. (2) Let L = {1, b,…

A: set of symbols = {a,b,c}

Q: State whether each of the following functions is injective, surjective, bijective, or none of these…

A: For convenience, let's say f: {1, 2} {a, b, c}. To define f, we need to determine f(1) and f(2).…

Q: For any string w=w1w2…wn, the reverse of w, written wR, is the string w in reverse order, wn…w2w1.…

A: Answer is given below-

Q: Suppose f : A → B is an injective function. Show that there is a function g : B → A such that g ◦ f…

A: An injective function, also known as a one-to-one function, is a type of function in…

Q: The mapping f: R → R, f(x) = x2, which of the following are correct? f is one-to-one. f is not a…

A:

Q: Consider the sets A={1,2,3} , B={a,b,c} and a function f:A→B defined as…

A:

Q: For any string w over ∑, writing its individual symbols so that w = w1w2…wn, we define the reverse…

A: Given: w = w1w2…wn with wi ∈ ∑ for 1 ≤ i ≤ n, Reverse of w is denoted by wR . wR = wn…w2w1 Let A be…

Q: Consider the following languages over the alphabet Σ = {a, b}: 1. L1 = {xwx : x, w ∈ Σ∗} 2. L2 =…

A: Making the distinction between regular and non-regular languages is a crucial task in the field of…

Q: Problem 1 Given an alphabet Σ, a string constructed from the alphabet is a concatenation of symbols…

A: This question is from regular expression:

Q: Prove or disprove whether the language P = {w^i | i ∈ N, w ∈ {a, b} ∗} is regular .

A: DFA

Q: Consider the language L = AEqB = {x ∈ {a, b}∗ | na(x) = nb(x)}. a. Show that if na(x) − nb(x) =…

A: Answer: I have given answered in the handwritten format in brief explanation

Q: For any string w over ∑, writing its individual symbols so that w = w1w2…wn, we define the reverse…

A: GIVEN: Let w be a string such that w=w1w2….wn. The reverse of w is wR. Where wR = wn……w3w2w1.   TO…

Q: justify whether each of the following functions is injective, surjective, bijective, or none of…

A: following function is surjective

Q: Let L be the set of exactly those strings over the alphabet {a, b, c, g} whose last symbol is either…

A: Given NFA should accept all strings over {a,b,c,g} even length Accepting strings are aa ab abcg aabb…

Q: Suppose I is a non empty regular language, over Σ, such that every w E L satisfies [w] = k, for some…

A: Given that, non-empty regular language L over the alphabet Σ such that every string w in L satisfies…

Q: Consider the language L = {a'b'c* | i # j,j # k, i # k}; it is not a context- free language. Show…

A: Pumping lemma for CFL is used to prove the language is not cfl using some valid string s.  Never use…

Q: Consider the following languages over the alphabet Σ = {a,b, c} L₁ = {a¹ b¹ck: i, j, k € N such that…

A: The pumping lemma for regular languages can be defined in such a way that it is a crucial end result…

Q: Let L be a strict subset of LAc (i.e. L c LACC). Then L is ( always / sometimes / never ) decidable.

A: Defined the given statement always, sometimes or never

Q: A palindrome is a string that reads the same backward as forward, i.e., w = wR. A palindromic…

A: Hey there, I am writing the required solution of the questin mentioned above. Please do find the…

Q: (a) L = {all words with exactly one b}. (b) L = {all words with at least one a}. (c) L = {all words…

A: For the alphabet Σ ={a, b}, construct an FA that accepts the following languages.(a) L = {all words…

Q: If A is any language, let A 1/3 – 1/3 be the set of all strings in A with their middle thirds…

A: the answer is given below:-

Q: Let L be the set of exactly those strings over the alphabet {a, b, c, g} whose last symbol is either…

A: Transition function :  that takes as arguments a state and an input symbol and returns a state.…

Q: Show that the language {w | numa(w) ≤ numb(w) ≤ numc(w) and w {a,b,c}*} is not ∈ context-free, where…

A: the answer is given below:-