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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- Expand on the concept of Black Box and then go into detail on primivitive functions: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) = aHere 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?Prove:Church’s thesis: The class of computable functions is equal to the class of intuitive computable functions.
- 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?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
- ) Consider the following functions. Decide whether these functions are injective,surjective, and invertible. Justify your answer (e.g., if you claim that a function is invertible, you need togive a justification as to why you think that function is invertible). Give counterexamples when needed.You can draw arrow diagrams to help justifying your answer.a) Function f: ℤ × ℤ → ℤ is defined as f((a, b)) = 2b – 4a.b) A = {1, 2, 3}. Function f: ?(A) → {0, 1, 2, 3} is defined as f(X) = |X| where |X| = size of X. Forexample, |{1, 2}| = 2. ?(A) is the power set of A.c) Function f: {0, 1}3 → {0, 1}3 is defined by the following rule. For each string s ∈ {0, 1}3,f(s) = f(x1x2x3) = x3x1x2, where x1, x2, x3 ∈ {0, 1}. For example, if x1 = a, x2 = b, and x3 = c, thenf(abc) = cab. Another example: f(011) = 101.7. 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.Directions: Let A = {1, 2, . . . , 9, 10}. Consider each of the following sentences. If it is a statement, then determine its truth value. If it is a propositional function, determine its truth set.