Given g = {(1,c),(2,a),(3,d)}, a function from X = {1,2,3} to Y = {a,b,c,d}, and f = {(a,r),(b,p),(c,δ),(d,r)}, a function from Y to Z = {p, β, r, δ}, write f o g as a set of ordered pairs.
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Given g = {(1,c),(2,a),(3,d)}, a function from X = {1,2,3} to Y = {a,b,c,d}, and f = {(a,r),(b,p),(c,δ),(d,r)}, a function from Y to Z = {p, β, r, δ}, write f o g as a set of ordered pairs.
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- The question describes a function S(k) which is defined as the sum of the positive divisors of a positive integer k, minus k itself. The function S(1) is defined as 1, and for any positive integer k greater than 1, S(k) is calculated as S(k) = σ(k) - k, where σ(k) is the sum of all positive divisors of k. Some examples of S(k) are given: S(1) = 1 S(2) = 1 S(3) = 1 S(4) = 3 S(5) = 1 S(6) = 6 S(7) = 1 S(8) = 7 S(9) = 4 The question then introduces a recursive sequence a_n with the following rules: a_1 = 12 For n ≥ 2, a_n = S(a_(n-1)) Part (a) of the question asks to calculate the values of a_2, a_3, a_4, a_5, a_6, a_7, and a_8 for the sequence. Part (b) modifies the sequence to start with a_1 = k, where k is any positive integer, and the same recursion formula applies: for n ≥ 2, a_n = S(a_(n-1)). The question notes that for many choices of k, the sequence a_n will eventually reach and remain at 1, but this is not always the case. It asks to find, with an explanation, two specific…Let l be a line in the x-yplane. If l is a vertical line, its equation is x = a for some real number a. Suppose l is not a vertical line and its slope is m. Then the equation of l is y = mx + b, where b is the y-intercept. If l passes through the point (x₀, y₀), the equation of l can be written as y - y₀ = m(x - x₀). If (x₁, y₁) and (x₂, y₂) are two points in the x-y plane and x₁ ≠ x₂, the slope of line passing through these points is m = (y₂ - y₁)/(x₂ - x₁). Instructions Write a program that prompts the user for two points in the x-y plane. Input should be entered in the following order: Input x₁ Input y₁ Input x₂The binomial coefficient C(N,k) can be defined recursively as follows: C(N,0) = 1, C(N,N) = 1, and for 0 < k < N, C(N,k) = C(N-1,k) + C(N - 1,k - 1). Write a function and give an analysis of the running time to compute the binomial coefficients as follows: A. The function is written using dynamic programming.
- Give an example of a function from Z to N that is one-to-one, but not onto.Simplify the complement of the following function: F(A,B,C,D)=(0,2,4,5,8,9,10,11) Your answer: F=((A'B'D)' (BC)'(AB)')' F=((A'BD)'(BC)'(AB)')' F=((A'B'D)'(B'C)'(AB)') F=((A'B'D')' (BC)'(AB)')1. Given the following data set: X: 0 2 4 6 9 11 12 15 17 19 22 25 Y: 5 6 7 6 9 8 7 10 12 12 15 10 (a) Write a Python program to fit a curve using: Polynomial function of order 4 Cubic spline fit Plot both the data set and the approximating curve for the polynomial and spline functions. (b) In the Python program, add labels along the x and y axes, and put a title of your choice for each plot.
- Let A = {c, n, b}, B = {x, y} and C = {0, 1}. Find A)A X B X C B)C X B X A C)B X C X CThe binomial coefficient C(N,k) can be defined recursively as follows: C(N,0) = 1, C(N,N) = 1, and for 0 < k < N, C(N,k) = C(N-1,k) + C(N - 1,k - 1). Write a function and give an analysis of the running time to compute the binomial coefficients as follows: A. The function is written recursively.Define a recursive function (rem r b) that, given a regular expression r and a bool b, returns a new regular expression r′ that matches exactly the set of all strings s such that string bs is matched by r. We will call r′ the remainder of r after division by b. For example, if r matches {T,FF T,TFF} and b = T, then r′ can be any regular expression that matches exactly the set {ε,FF} (because T and TFF are the only strings matched by r that begin with b = T, and their remainders are ε and FF, respectively). Here are some examples of what your function could output (but these are not the only answers!): • rem (false(false+true)∗) false = (false+true)∗ • rem (false(false+true)∗) true = ∅• rem (false∗ + true∗) true = true∗• rem ((false∗)(true∗)) true = true∗ Your implementation need not output these exact regular expressions as long as it always outputs an equivalent regular expression (i.e., one that matches the same set of strings as the given answer). These are also not the only test…
- Although the plot function is designed primarily for plotting standard xy graphs, it can be adapted for other kinds of plotting as well. b. Make a plot of the curve, which is defined parametrically by the equations x = 2cosθ + cos2θ, y = 2sinθ - sin2θ, where 0 < θ < 2π. Take a set of values of θ between zero and 2π and calculate x and y for each from the equations above, then plot y as a function of x. b. Taking this approach a step further, one can make a polar plot r = f(θ) for some function f by calculating r for a range of values of θ and then converting r and θ to Cartesian coordinates using the standard equations x = r cosθ, y = r sinθ. Use this method to make a plot of the function r = ecosθ – 2 cos(4θ) + sin5 (θ/12) in the range 0 <= θ <= 24π. use python code to answer the highlight oneDefine a function S : Z+ → Z+ as follows. For each positive integer n, S(n) =the sum of the positive divisors of n. S (7) ?Consider the following for loops in R. For each for loop, list the values (in order) that the variable i takes on in the body of the loop. Briefly (in no more than a few sentences) explain why. a) for(i in 1+2:3.4*5) { } b) for(i in dim(matrix(0, nr = 7, nc = 8))) { } c) for(i in rnorm(3)) { } d) for(i in iris[1:3,3]) { } e) for(j in c(1, 2, 3, 4, 5)) { } f) for(i in (function(x) x*x)(c(1, 2, 3))) { } g) for(i in NULL) { } h) for(i in strsplit(as.character(4*atan(1)),’’) [[1]][1:10]) { }