Problem 1. Prove that the following functions are Primitive Recursive. I – 1 if x > 0, (1) mPred(x) = for r € N. if x = 0. [1 if x > 0, 0if x = 0. (2) sgn(x) = for x E N. %3D if x > 0, (3) sgn(x) = for x Ν. 1 if x = 0.
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- es remaining 8. Consider the function f:NxN-N defined recursively by: 1) Base case: Let meN and define (0,m) = 0 2) Recursive case: For any x,meN, x>0, define f(x,m) = (x-1,m) + (m+m) Prove the following theorem holds using proof by induction: Thereom: For any n,meN, m>0 we have (n.m) I m = n+n Fill in your answer here 9 Help BIU X, x L - ɔE =N E X Format1. For the function defined recursively by f(0)=5 and f(n)=4f (n-1)+3, answer the following: a. Find a closed form representation for this function. Your closed form should not include any series. b. Prove that your representation is correct using a formal inductive argument.Determine whether the proposed definition isa valid recursive definition of a function f from the setof nonnegative integers to the set of integers. If f is welldefined, find a formula for f(n) when n is a nonnegativeinteger and prove that your formula is valid. f(0) = 1, f(n) = −f(n − 1) for n ≥ 1
- Determine whether the proposed definition isa valid recursive definition of a function f from the setof nonnegative integers to the set of integers. If f is welldefined, find a formula for f(n) when n is a nonnegativeinteger and prove that your formula is valid. Do NOT use proofs. f(0) = 1, f(n) = −f(n − 1) for n ≥ 14. Find a closed form representation for the function defined recursively by f(0)=5 and f(n+1)=3f(n)+4. Prove that your representation is correct using an inductive argument.Given A = {1,2,3} and B={u,v}, determine. a. A X B b. B X B
- Given g = {(1,c),(2,a),(3,d)}, a function from X = {1,2,3} to Y = {a,b,c,d}, and f = {(a,r),(b,p),(c,δ),(d,r)}, a function from Y to Z = {p, β, r, δ}, write f o g as a set of ordered pairs.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.Let p be a proposition and P be a propositional function. Identify if the following statement is always true, never true, or only sometimes true/false: T→p⇒T^p Always True O Never True O Sometimes True Suppose there is a robot that builds a copy of itself in 2 hours. The copy then starts to build copies of itself as well. Let be the total number of fully functional robots after n hours. Suppose ro = 1. Recursive Case, 'n = 'n = [(n-2) +2 In = [(n-2) *2 In = [(n-1) +1 O None. Function description: g: Z→R g(x) = (x − 2)(x+2)x Identify if g is: 1. One-to-One II. Onto III. One-to-One Correspondence O IV. None
- 7. Find a closed form representation for the function defined recursively by f(1)-10 and f(n)=5ƒ(%) + n.Let f (n) and g(n) be functions with domain {1, 2, 3, . . .}. Prove the following: If f(n) = O(g(n)), then g(n) = Ω(f(n)).If S = { x | 0 ≤ x ≤ 10}, A = { x | 1 ≤ x ≤ 5}, B = { x | 1 ≤ x ≤ 6}, and C = { x | 2 ≤ x ≤ 7}(a) S ⋃ C(b) A ⋃ B(d) A’ ⋂ C(c) A’⋃ (B ⋂ C)(e) (A ⋂ B) ⋃ (B ⋂ C) ⋃ (C ⋂ A)