Assume that we are given M=R e^+and the function on the...

Assume that we are given $M=R e^{+}$and the function on the positive reals given by $F(x)=x / 2$. Use $F$ to write the recursive definition of a function $r: N a t \rightarrow \operatorname{Re}^{+}$satisfying $r(0)=1$. Also, compute the values of $r(k)$ for $$ k=s(0), \quad s(s(0)), \quad s(s(s(0))), \quad s(s(s(s(0)))), \quad s(s(s(s(s(0))))) . $$

Assume that we are given $M=R e^{+}$and the function on the positive reals given by $F(x)=x / 2$. Use $F$ to write the recursive definition of a function $r: N a t \rightarrow \operatorname{Re}^{+}$satisfying $r(0)=1$. Also, compute the values of $r(k)$ for
$$
k=s(0), \quad s(s(0)), \quad s(s(s(0))), \quad s(s(s(s(0)))), \quad s(s(s(s(s(0))))) .
$$

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Key Concepts

Base Case

The base case provides the starting point of a recursive definition. It is the explicitly given value of the function at the initial argument (often 0 or 1). The base case is critical because it anchors the recursion and ensures that the recursive process is well-defined.

Recursive Step

The recursive step is the rule that defines the function for inputs greater than the base case. It uses the value of the function at a smaller input to determine the value at the next input. This method, through repeated application, generates the sequence or function values across the domain.

Successor Function

In recursive definitions on the natural numbers, the successor function (often denoted as s(n) or similar) represents the process of moving from an element to the next one. It serves as a formal notation to indicate the next natural number, making it useful in defining functions recursively over the naturals.

Recursive Definition

A recursive definition defines an object in terms of itself. In mathematics, especially in the context of functions, this means establishing a base case that gives an initial value and then a rule (called the recursive step) that tells you how to obtain the value for any subsequent input from the value at a previous point. This approach is essential for defining sequences and functions on natural numbers.

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