For each of the following functions, determine whether the function is: Injective (one-to-one). Surjective (onto). ⚫ Bijective. Justify your answers. 1.3 f: R→ ZZ such that f(x) = [4x] (i.e., ceiling of 4x).
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- Prove:Church’s thesis: The class of computable functions is equal to the class of intuitive computable functions.Q. Let A = {a, b, c, d, e} and B = {1, 2, 3, 4, 5, 6, 7, 8}. How many functions f : A → B(a) ... are injective?(b) ... are not injective?(c) ... are such that f(a) = f(b) = f(c)?(d) ... are such that exactly three elements of A have 8 as an image?(e) ... are surjective?Determine if each function is injective, surjective or bijective. Give one counterexample for each that it is not. 1. function f from {a,b,c,d} to itself, where f(a) = d, f(b) = b, f(c) = a, f(d) = c
- 7. Let P denote the set of all phones in the world such that p ∈ P is a phone. Thus, S(p)denotes that “p is a SamsungTM phone”, N(p) denote that “p is a NokiaTM phone”, and G(p)denote that “p is a GoogleTM phone”. Therefore, express each of the following statements usingquantifiers, logical operations, and the propositional functions: S(p),N(p),G(p).(a) There is a GoogleTM phone that is also a SamsungTM phone.(b) Every NokiaTM phone is a GoogleTM phone.(c) No NokiaTM phone is a GoogleTM phone.(d) Some NokiaTM phones are also SamsungTM phones.(e) Some NokiaTM phones are also GoogleTM phones and some are not. PS: Please do not answer them in a paper format.pick multiple answers on the second one !For the first problem, in parts b and c, how are the variables being taken from functions that they are not defined in? To clarify, in part c, how is the variable "b" taken from fun2 when it is not defined in that in function. Shouldn't the answer there just be "b(fun1)"
- Create a geometric series to model a geometric series. using friend functions overload the ~ operator to find tbe nth term of the series. likewise, overload the"! " operator using a friend function to find the sum of geometric series.Let A = {a, b, c, d, e} and B = {1, 2, 3, 4, 5, 6, 7, 8}. How many functions f : A → B(a) ... are injective?(b) ... are not injective?(c)... are such that f(a) = f(b) = f(c)?(d) ... are such that exactly three elements of A have 8 as an image?(e) ... are surjective?Let A and B be sets. How to show the following?:1. If A es is countable and there exists a surjective function f: A → B, then B is countable.2. If B is countable and there exists an injective function f: A → B, then A is countable
- Answer the following multiple choice questions: - If B represented the set of Boolean values true and false, then the set of Boolean functions B x B → B is... A) finite B) uncountably infinite C) countably infinite - The set of functions from {0,1} to N is... A) finite B) uncountably infinite C) countably infinite - The set of functions from N to {0,1} is... A) finite B) uncountably infinite C) countably infiniteCount the total number of different one-to-one functions from the set {0, 1} into the set {1, 2, 3,...,n}, where n is a positive integer.Example 2: Let f and g be functions from the set of integers to the set of integers defined by f (x)= 2x+3 and g(x)= 3x+2. What is the composition of f and g, and also the composition of g and f?