Determine whether or not the function f : Z × Z ! Z is onto, if f((m, n)) =m-n.
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A: GIVEN:
Q: function
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Q: K map
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Determine whether or not the function f : Z × Z ! Z is onto, if f((m, n)) =m-n.
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- How do we define that a function f(n) has an upper bound g(n), i.e., f(n) ∈ O(g(n))?Give an example of a function from Z to N that is one-to-one, but not onto.Order the following functions by asymptotics with respect to Ω. That is, find an orderingf1 , f2 , · · ·, of the following functions such that f1 = Ω(f2), f2 = Ω(f3) and so on.n2, (√2)logn , n!, log(n)!, (3/2)n , n3, log2n, loglogn, 4logn, 2n, nlogn, 2logn , 2√2logn , log(n!)
- Give an explicit formula for a function from the set of all integers tothe set of positive integers that is onto but is not one-to-one.How many lines will the following function write? Write the recurrence relation of each and solve using the Master theorem. Give your answer as a function of n (in the form Θ( · )).Write a function linear_independence that takes a collection of vectors with integer entries (each written as a list), and returns True if this collection of vectors is linearly independent, and False otherwise. Examples: linear_independence([1,2]) should return True. linear_independence([1,3,7],[2,8,3],[7,8,1]) should returnTrue. linear_independence([1,3,7],[2,8,3],[7,8,1],[1,2,3]) should return False.
- Please help me with these question. SHow all you work. Thank you 1. Prove that∀k ∈ N, 1k + 2k + · · · + nk ∈ Θ(nk+1). 2. Suppose that the functions f1, f2, g1, g2 : N → R≥0 are such that f1 ∈ Θ(g1) and f2 ∈ Θ(g2).Prove that (f1 + f2) ∈ Θ(max{g1, g2}).Here (f1 + f2)(n) = f1(n) + f2(n) and max{g1, g2}(n) = max{g1(n), g2(n)}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 --CashOrNothingOption = function(S, k, Time, r, q, sigma) {d1 = (log(S / k) + (r - q + sigma ^ 2 / 2) * Time) / (sigma * sqrt(Time))d2 = d1 - sigma * sqrt(Time)if(k>S){return (k * exp (-r * Time) * (-d2))}elseprint("zero")}
- For this question, you will be required to use the binary search to find the root of some function f(x)f(x) on the domain x∈[a,b]x∈[a,b] by continuously bisecting the domain. In our case, the root of the function can be defined as the x-values where the function will return 0, i.e. f(x)=0f(x)=0 For example, for the function: f(x)=sin2(x)x2−2f(x)=sin2(x)x2−2 on the domain [0,2][0,2], the root can be found at x≈1.43x≈1.43 Constraints Stopping criteria: ∣∣f(root)∣∣<0.0001|f(root)|<0.0001 or you reach a maximum of 1000 iterations. Round your answer to two decimal places. Function specifications Argument(s): f (function) →→ mathematical expression in the form of a lambda function. domain (tuple) →→ the domain of the function given a set of two integers. MAX (int) →→ the maximum number of iterations that will be performed by the function. Return: root (float) →→ return the root (rounded to two decimals) of the given function. START FUNCTION def binary_search(f,domain, MAX =…For this question, you will be required to use the binary search to find the root of some function f(x)f(x) on the domain x∈[a,b]x∈[a,b] by continuously bisecting the domain. In our case, the root of the function can be defined as the x-values where the function will return 0, i.e. f(x)=0f(x)=0 For example, for the function: f(x)=sin2(x)x2−2f(x)=sin2(x)x2−2 on the domain [0,2][0,2], the root can be found at x≈1.43x≈1.43 Constraints Stopping criteria: ∣∣f(root)∣∣<0.0001|f(root)|<0.0001 or you reach a maximum of 1000 iterations. Round your answer to two decimal places. Function specifications Argument(s): f (function) →→ mathematical expression in the form of a lambda function. domain (tuple) →→ the domain of the function given a set of two integers. MAX (int) →→ the maximum number of iterations that will be performed by the function. Return: root (float) →→ return the root (rounded to two decimals) of the given function. START FUNCTION def binary_search(f,domain, MAX =…Show that if f(x) and g(x) are functions from the set of real numbers to the set of real numbers, then f(x) is O(g(x)) if and only if g(x) is Ω(f(x)).