Consider a function f : S→T defined by f(n) = n², where the domain is the set of consecutive integers S = {-7,..., -2, -1, 0, 1, 2, ..., 7} and the range is the set of consecutive squares T = {-64,..., -9,–4, −1, 0, 1, 4, 9, . . . ,64}. Which one of the following choices correct describes f? Oone-to-one, but NOT onto Oonto, but NOT one-to-one O one-to-one and onto Oneither one-to-one nor onto Onone of these
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- 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.A = {0, 1, 2, 3, 4} B = {2, 3, 4, 5}Given the sets A and B, how many constants maps are there from A into B.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 x₂
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- give set of R(x): x < x^2 where the domain is ZGiven a function f : A → B, we get a corresponding function f : P(A) → P(B), that maps a subset X ⊆ A to its image f(X) ⊆ B. What condition must f : A → B satisfy for f : P(A) →P(B) to be one-to-one? What condition must f : A → B satisfy for f : P(A) → P(B) to be onto?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…
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