Show that each of these proposed recursive definitions of a...

Show that each of these proposed recursive definitions of a function on the set of positive integers does not produce a well-defined function. a) $F(n)=1+F(\lfloor(n+1) / 2\rfloor) \quad$ for $\quad n \geq 1 \quad$ and $F(1)=1 .$ b) $F(n)=1+F(n-2)$ for $n \geq 2$ and $F(1)=0$ c) $F(n)=1+F(n / 3)$ for $n \geq 3, F(1)=1, F(2)=2,$ and $F(3)=3$ d) $F(n)=1+F(n / 2)$ if $n$ is even and $n \geq 2, F(n)=1+$ $F(n-2)$ if $n$ is odd, and $F(1)=1 .$ e) $F(n)=1+F(F(n-1))$ if $n \geq 2$ and $F(1)=2$

Show that each of these proposed recursive definitions of a function on the set of positive integers does not produce a well-defined function.
a) $F(n)=1+F(\lfloor(n+1) / 2\rfloor) \quad$ for $\quad n \geq 1 \quad$ and $F(1)=1 .$
b) $F(n)=1+F(n-2)$ for $n \geq 2$ and $F(1)=0$
c) $F(n)=1+F(n / 3)$ for $n \geq 3, F(1)=1, F(2)=2,$ and $F(3)=3$
d) $F(n)=1+F(n / 2)$ if $n$ is even and $n \geq 2, F(n)=1+$ $F(n-2)$ if $n$ is odd, and $F(1)=1 .$
e) $F(n)=1+F(F(n-1))$ if $n \geq 2$ and $F(1)=2$

Key Concepts

Termination of Recursion

For a recursive function to be well-defined, every chain of recursive calls must eventually converge to a base case. This requires that each recursive step moves closer to the base case, ensuring that the recursion does not continue indefinitely. If the recursive process can cycle, oscillate, or otherwise fail to reach the base case, the function remains undefined for some inputs.

Domain Considerations in Recursion

The recursive definition must ensure that the transformation applied to the input during a recursive call always yields an argument that lies within the function's domain. If a recursive rule can produce values outside the intended domain or create ambiguities, then the function may not provide a unique output for each input, rendering it not well-defined.

Base Case

The base case in a recursive definition provides the starting point or fixed value for the smallest input(s), which does not rely on any further recursive calls. It serves as the anchor to which all other recursive instances eventually reduce. Without an appropriate base case, or if the base case is mismatched with the recursion, the function may not be defined for all inputs.

Well-defined Function

A function is well-defined if every element in its domain is assigned exactly one value from its codomain. In the context of recursive definitions, the function must produce a consistent and unique result for every input, meaning that the recursion has a clear starting point and a guaranteed path that eventually terminates at the base case.

Recursive Definition

A recursive definition specifies the value of a function for larger inputs in terms of its value on smaller inputs. It typically consists of one or more rules or formulas that refer back to the function itself. Such definitions require careful formulation so that every instance of the function can eventually be reduced to a base case, ensuring that each input is assigned a unique output.

Transcript

00:02 We are given recursive definitions of functions.

00:06 We are asked to prove that these do not produce well -defined functions.

00:17 In part a, we're given that f of n is equal to 1 plus f of the floor of n plus 1 over 2.

00:46 For n greater than are equal to 1, and we're given that f of 1 is equal to 1.

00:59 So we have on the one hand f of 1 is defined to be 1, but since 1 is greater than are equal to 1, it also follows that f of 1 is equal to 1 plus f of the floor of 1 plus 1 over 2, which is the floor of 2 over 2 or the floor of 1, which is just equal to 1 plus 1 plus f of 1.

01:38 And so we get that 1 is equal to 0, which is clearly a contradiction.

01:49 So it follows that f of 1 is not well defined.

01:55 And so it follows that f is not well defined.

02:07 In part b, we're given that f of n is equal to 1 plus f of n minus 2 for n greater than or equal to 2.

02:24 And that f of 1 is equal to 0.

02:50 Now notice that f of 2, since it's greater than equal to 2, 2 is going to be 1 plus f of 2 minus 2, which is 0.

03:08 F of 0 does not exist since f is only defined for the positive integers.

03:19 So f of 2 is not well defined.

03:27 So f is not well defined.

03:37 In part c, we're given that f of n is equal to f1 plus f of n over 3, for n greater than are equal to 3.

03:56 F of 1 is equal to 1...