Question

Are the following statements true or false?

1. Let a be the sequence of numbers defined by the rules a₀ = 0 and, for any n, a_n+1 = (n + 1) - a_n. Then for any natural n, a_n is the natural denoted in Java by "n/2".

2. Let f be any function from naturals to naturals and let g(n) be the sum, for i from 1 to n, of f(i). Suppose I have a function h from naturals to naturals that satisfies the property that for any natural n, h(n+1) = h(n) + f(n+1). Then g and h must be the same function.

3. Let the function z(n), from naturals to integers, be defined by the rules z(0) = 1, and for any natural n, z(2n) = z(n)² and z(2n+1) = -z(n)². Then we can prove by strong induction that for any natural n, z(n) = (-1)ⁿ.

Answer 1

1. Let a be the sequence of numbers defined by the rules a₀ = 0 and, for any n, a_n+1 = (n + 1) - a_n. Then for any natural n, a_n is the natural denoted in Java by "n/2".

Answer:

a₁ = 2 - a₀ = 2

a₂ = 3 - a₁ = 1

a₃ = 4 - 1 = 3

Clearly, a_n != n/2, hence the statement is false.

2. Let f be any function from naturals to naturals and let g(n) be the sum, for i from 1 to n, of f(i). Suppose I have a function h from naturals to naturals that satisfies the property that for any natural n, h(n+1) = h(n) + f(n+1). Then g and h must be the same function.

This statement is true.

h(1) = h(0) + f(1) = f(1) considering h(0) = 0

h(2) = h(1) + f(2) = f(1) + f(2)

h(3) = h(2) + f(3) = f(1) + f(2) + f(3) and so on.

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