Computer Science Exercise 4. Prove the following properties of regular expressions a) ∗= Λ b) R∗|R2= R∗
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Computer Science
Exercise 4. Prove the following properties of regular expressions
a) ∗= Λ
b) R∗|R2= R∗
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- Let α = (a + b)∗ab(a + b)∗. Give a regular expression equivalent to ∼ α in the following cases.Simplify the expressions as much as possible (either using the rules from the lecture or othermeans).(a) Σ = {a,b}(b) Σ = {a,b,c}Automata and ComputabilityDiscrete mathematics applied to computer science: homework question. Let f : A → A be a function in the domain A =/= ∅. Confirm or deny, with justification: f is a bijection if and only if the composition f ◦ f is a bijection. I don't understand how I should approach this problem. Thank you for your gracious help.Show the function f(i; k) is primitive recursive where f(i; k) = Pi.Pi+1............Pi+k. Recall, Pn is the nth prime number and P0 = 0.
- Find closed form representation for recursively defined function: T(n) = 7, when n = 1 T(n) = 5T(n/4) + n^2, when n > 1. Explore power of two. Find closed form representation in the form where T(n) = A*n^B + C*n^D. Find A, B, C, and D. Hint: T(4) = ?, T(4^2) = ?, T(4^3) = ? to find the patten.Computer Science Suppose L ⊆{a, b}∗ is defined as follows: Λ∈L; for every x and y in L, the strings axb, bxa, and x yare in L. Show that L =AEqB , the language of all strings x in {a, b}∗ satisfying n_a (x) =n_b (x).Exercise 1.A set W of strings of symbols is defined recursively by a, b, and d belong to W. If x belongs to W, so does a(x)d. Provide 3 strings made of 3 letters each belonging to W and show the procedure used to obtain them. Provide 3 strings made of at least 5 letters each belonging to W and show the procedure used to obtain them.
- Develop the desktop proof of the following recursive exercise with V = [3,5,2,7] and num = 4 (Logically without the use of programs)A prime number is a natural number greater than 1 which is not a product of two smaller natural numbers. Prove or disprove: For every integer q, if q > 7, then q can be written as q = (a + b * c) such that the following properties hold a and b are prime numbers, c is an integer greater than 1.S is a set of strings over the alphabet {a, b}* recursively defined as: Base case: λ ∈ S, a ∈ S, b ∈ S Recursive rules: If x ∈ S, then Rule 1: axb ∈ S Rule 2: bxa ∈ S List all the strings in S of length 3.
- For the following pairs of expressions, find a unifier for each pair if a unifier exists. If a unifierdoes not exist, explain why.(a) p(X,Y) and p(a,Z)(b) p(X,X) and p(a,b)(c) f(X,Y) and f(a,g(a))(d) q(X) and ¬q(a)9. Given the regular expression r = a(a+b)*b a) Show an nfa that accepts the language L(r) b) Show an npda that accepts the language L(r)Give a recursive definition for an (where n=1,2,3,...) if an=2n+5