For each of the following functions, determine whether the function is: Injective (one-to-one). Surjective (onto). ⚫ Bijective. Justify your answers. 1.1 f:Z+ → Z+ such that f(x) = 5+x.
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- Which functions are one-to-one? Which functions are onto? Describe the inversefunction for any bijective function.(a) f : Z → N where f is defined by f (x) = x4 + 1(b) f : N → N where f is defined by f (x) = { x/2 if x is even, x + 1 if x is odd}(c) f : N → N where f is defined by f (x) = { x + 1 if x is even, x − 1 if x is odd}Ql: The Collatz conjecture function is defined for a positive integer m as follows. (COO1) g(m) = 3m+1 if m is odd = m/2 if m is even =1 if m=1 The repeated application of the Collatz conjecture function, as follows: g(n), g(g(n)), g(g(g(n))), ... e.g. If m=17, the sequence is 1. g(17) = 52 2. g(52) = 26 3. g(26) = 13 4. g(13) = 40 5. g(40) = 20 6. g(20) = 10 7. g(10) = 5 8. g(5) = 16 9. g(16) = 8 10. g(8) = 4 11. g(4) = 2 12. g(2) = 1 Thus if m=17, apply the function 12 times in order to reach m=1. Use Recursive Function.Determine whether each of the following functions f : {a,b,c,d} -> {a,b,c,d} is one-to-one and/or onto. (a) f(a) = b, f(b) = a, f(c) = b, f(d) = c (b) f(a) = b, f(b) = b, f(c) = d, f(d) = c (c) f(a) = b, f(b) = a, f(c) = c, f(d) = d (d) f(a) = d, f(b) = a, f(c) = c, f(d) = b (e) f(a) = c, f(b) = d, f(c) = a
- Prove:Church’s thesis: The class of computable functions is equal to the class of intuitive computable functions.Here are the definitions for functions f and g. f R R defined by f(x) = x2 g: R R defined by g(x) = x3 Answer the following questions 1) is fa 1-1 function? is fa onto function? is f a bijection? Does f have an inverse? If yes, what is the inverse of f? 1) is ga 1-1 function? is g an onto function? is g a bijection? Does g have an inverse? If yes, what is the inverse of g?Let A = {1, 2, 3, 4} and B = {a, b, c}. Give an example of a function f: A -> B that is neither injective nor surjective.
- 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?Please written by computer sourceDetermine 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
- Answer question number four please.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)"