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This is an Automata and Computation question on Functions/Sets.
A function is relationship between input and output having condition that for each input there is exactly one output.
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- Suppose f : A → B is an injective function. Show that there is a function g : B → A such thatg ◦ f = idA. Here, idA : A → A is the identity function on A, i.e., the function that satisfies idA(a) = afor all a ∈ A. (Assume f is non-trivial, i.e., A ̸= ∅!)is this functions from R to R is a bijection or not f(x)=(x2+2)/(x2+3)Discrete 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.
- For function f from {a, b, c, d} to {1, 2, 3, 4, 5}. f(a) = 4, f(b) = 5, f(c) = 1 and f(d) = 3, The set containing the domain has the following elementsGive 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.A function is "almost onto" if it misses exactly one element of its codomain. How many almost onto functions are there from [k] to [n]?
- Prove:Church’s thesis: The class of computable functions is equal to the class of intuitive computable functions.State whether each of the following functions is injective, surjective, bijective, or none ofthese categories: (N is the set of natural numbers, and Σ∗is the set of all strings over Σ.)(a) f : N → N , f(n) = n div 3(b) g : N → N , g(n) = n2 + 2n + 1 (c) h : Σ∗ → Σ∗, where Σ = {a, b}, and h(w) = a|w| (d) h : Σ∗ → Σ∗, where Σ = {a, b}, and h(w) = wRState whether each of the following functions is injective, surjective, bijective, or none ofthese categories: (N is the set of natural numbers, and Σ∗is the set of all strings over Σ.)(a) f : N → N , f(n) = n div 3(b) g : N → N , g(n) = n2 + 2n + 1(c) h : Σ∗ → Σ∗, where Σ = {a, b}, and h(w) = a|w|. (d) h : Σ∗ → Σ∗, where Σ = {a, b}, and h(w) = wR.
- The function s from is a(n)… a. Injection b. Surjection c. Bijection d. Function that isn't an injection or a surjectionWrite 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.Find a closed form representation for the following recursively defined function. Give the run-time complexity of each recursively defined function.