Prove that the function y = n* + 3 can't belong or does not belong to the set of 0(1) using Big-0 Notation's formal definition. Explain your defense and proof thoroughly.
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- Show that a function y = n^4 + 3 can not belong to the set O(1) using the formal definition of Big-OHow do we define that a function f(n) has an upper bound g(n), i.e., f(n) ∈ O(g(n))?Using the pumping lemma, show {a^n b^m a^m b^n} is nonregular language. Show it by listing all possible cases of substring y and considering each of them using the pumping lemma. How many possible cases of substring y exist?
- Let f and g be the function from the set of integers to itself, defined by f(x) = 2x + 1 and g(x) = x ^2. The composition of fog (-2) is --For f (a, b) = (a | b) | b(a) Simplify f (a, b).(b) Find DNF for f (a, b).(c) Is f (a, b) satisfiable?Prove f(n) = n3 - (n2 log2 n) + 2n f(n) ∈ O(n3) by giving the constants (c, n0) and arguing that your constants hold as n goes to infinity by either "chaining up", "chaining down", carefully treating the inequality as an equality and doing some algebra, or even weak induction. Also, what is the minimum val4ue for c (in the definition of O() ) forj this problem and is it an exclusive or inclusive minimum/bound.
- Apply a suitable approach to compare the asymptotic order of growth forthe following pair of functions. Prove your answer and conclude by telling if f(n) ?Ѳ(g(n)), f(n) ?O(g(n)) or f(n) ?Ω(g(n)). f(n) = 100n2+ 20 AND g(n) = n + log nShow that (p ∧ q) → r and (p → r) ∧ (q → r) is logically equivalent.Suppose we have positive integers a, b, and c, such that that a and b are not relatively prime, but c is relatively prime to both a and b . Let n = s × a + t × b be some linear combination of a and b, where s and t are integers. Prove that n cannot be a divisor of c. Follow the definition of relative primes, and use contradiction.
- Determine whether the proposed definition isa valid recursive definition of a function f from the setof nonnegative integers to the set of integers. If f is welldefined, find a formula for f(n) when n is a nonnegativeinteger and prove that your formula is valid. Do NOT use proofs. f(0) = 1, f(n) = −f(n − 1) for n ≥ 1Match the following sentence to the best suitable answer: - A. B. C. D. for the linear congruence ax=1(mod m), x is the inverse of a, if__________ - A. B. C. D. What is -4 mod 9 ? - A. B. C. D. The solution exists for a congruence ax=b(mod m) such that GCD(a,m)=1 and - A. B. C. D. (107+22)mod 10 is equivalent to :_________ A. 5 B. c divides b C. GCD(a,m)=1 D. 9 mod 10Could you assist me with this issue? I'm finding it challenging to grasp the solution. Would you mind providing a step-by-step walkthrough along with a comprehensive explanation? question that I need help with: Show that P is closed under union, concatenation, and complement.