Question

In: Advanced Math

Suppose f : N→N satisestherecurrencerelation f(n + 1) (f(n) 2 if f(n)iseven 3f(n)+ 1 if f(n)isodd...

Suppose f : N→N satisestherecurrencerelation f(n + 1) (f(n) 2 if f(n)iseven 3f(n)+ 1 if f(n)isodd . Notethatwiththeinitialcondition f(0) 1,thevaluesofthefunction are: f(1) 4, f(2) 2, f(3) 1, f(4) 4, and so on, the images cyclingthroughthosethreenumbers. Thus f isNOTinjective(andalso certainlynotsurjective). Mightitbeunderotherinitialconditions?3 (a) If f satisestheinitialcondition f(0) 5,is f injective? Explain whyorgiveaspecicexampleoftwoelementsfromthedomain withthesameimage. (b) If f satisestheinitialcondition f(0) 3,is f injective? Explain whyorgiveaspecicexampleoftwoelementsfromthedomain withthesameimage. (c) If f satisestheinitialcondition f(0) 27,thenitturnsoutthat f(105) 10 and no two numbers less than 105 have the same image. Could f beinjective? Explain. (d) Prove that no matter what initial condition you choose, the functioncannotbesurjective.

Solutions

Expert Solution

Colby Messinger answered 2 years ago
Next > < Previous

Related Solutions

2. The Fibonacci sequence is defined as f(n) = f(n - 1) + f(n - 2)...
2. The Fibonacci sequence is defined as f(n) = f(n - 1) + f(n - 2) with f(0) = 0 and f(1) = 1. Find f(54) by a program or maually. Note that this number must be positive and f(53) = 53.......73 (starting with 53 and ending with 73). I must admit that my three machines including a desktop are unable to find f(54) and they quit during computation. The answer is f(54) = 86267571272 */ The Java code: public...
Fibonacci Sequence: F(0) = 1, F(1) = 2, F(n) = F(n − 1) + F(n −...
Fibonacci Sequence: F(0) = 1, F(1) = 2, F(n) = F(n − 1) + F(n − 2) for n ≥ 2 (a) Use strong induction to show that F(n) ≤ 2^n for all n ≥ 0. (b) The answer for (a) shows that F(n) is O(2^n). If we could also show that F(n) is Ω(2^n), that would mean that F(n) is Θ(2^n), and our order of growth would be F(n). This doesn’t turn out to be the case because F(n)...
a) write a program to compute and print n, n(f), f(f(n)), f(f(f(n))).........for 1<=n<=100, where f(n)=n/2 if...
a) write a program to compute and print n, n(f), f(f(n)), f(f(f(n))).........for 1<=n<=100, where f(n)=n/2 if n is even and f(n)=3n+1 if n is odd b) make a conjecture based on part a. use java
Suppose f(1) = 2, g(1) = −1, f′(1) = 3, g′(1) = 2, f(2) = 1,...
Suppose f(1) = 2, g(1) = −1, f′(1) = 3, g′(1) = 2, f(2) = 1, and f′(2) = 0.2. Calculate the derivatives of the following functions at the provided point (be careful with using the correct values) 3 pts each: (a) d/dx (e^xf(x)) when x=2 (b) d/dx (f(x)/g(x)) when x=1 (c) d/dx (ln(xf(x))) when x=1
1.)Prove that f(n) = O(g(n)), given: F(n) = 2n + 10; g(n) = n 2.)Show that...
1.)Prove that f(n) = O(g(n)), given: F(n) = 2n + 10; g(n) = n 2.)Show that 5n2 – 15n + 100 is Θ(n2 ) 3.)Is 5n2 O(n)?
Define the following function f(n) = 5(2^n)-(2^(n-1)), n ≥ 1. Write a recursive definition for the...
Define the following function f(n) = 5(2^n)-(2^(n-1)), n ≥ 1. Write a recursive definition for the function f(n)? Consider the following recurrence: an= 2an-1 + 3 (where a1 = 1). Compute the values of an for n ≤ 5. Find a solution for the recurrence definition and validate its correctness. Consider the following recurrence: an=2an-1 +an-1an-2 (where a1 = 1). Compute the values of an for n ≤ 5.
2. Let f(x) ≥ 0 on [1, 2] and suppose that f is integrable on [1,...
2. Let f(x) ≥ 0 on [1, 2] and suppose that f is integrable on [1, 2] with R 2 1 f(x)dx = 2 3 . Prove that f(x 2 ) is integrable on [1, √ 2] and √ 2 6 ≤ Z √ 2 1 f(x 2 )dx ≤ 1 3 .
For f: N x N -> N defined by f(m,n) = 2m-1(2n-1) a) Prove: f is...
For f: N x N -> N defined by f(m,n) = 2m-1(2n-1) a) Prove: f is 1-to-1 b) Prove: f is onto c) Prove {1, 2} x N is countable
If f(n) = 3n+2 and g(n) = n, then Prove that f(n) = O (g(n))
If f(n) = 3n+2 and g(n) = n, then Prove that f(n) = O (g(n))
proof: L t^(n+1)*f(t)=(-1)^(n+1)*(d^(n+1)/ds^(n+1))*F(s)
proof: L t^(n+1)*f(t)=(-1)^(n+1)*(d^(n+1)/ds^(n+1))*F(s)
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT