Give examples of total unary number-theoretic functions that...

Give examples of total unary number-theoretic functions that satisfy the following conditions.
a) $g$ is not $i d$ and $h$ is not $i d$ but $g \circ h=i d$
b) $g$ is not a constant function and $h$ is not a constant function but $g \circ h$ is a constant function

Key Concepts

Constant Functions

A constant function is one that returns the same value regardless of the input. In part (b) of the problem, even though neither g nor h is constant individually, their composition results in a constant function. Understanding the behavior and construction of constant functions is essential here in determining how a non-constant function composed with another can collapse all outputs to a single value.

Identity Function

The identity function, usually denoted id, maps every element in the domain to itself (i.e., id(x) = x). It is a pivotal concept in function composition because composing any function with the identity function leaves the other function unchanged. The challenge in these problems is to find non-identity functions whose composition still results in the identity function for part (a).

Inverse Functions

An inverse function reverses the effect of a function such that composing a function and its inverse yields the identity function. In part (a) of the problem, even though neither g nor h is the identity function, their composition is the identity, indicating that one is a kind of inverse of the other (in a one-sided sense). This highlights the concept of mutual cancellation even when the functions themselves are not trivial.

Total Unary Number?Theoretic Functions

These are functions defined on the set of natural numbers (or integers) that take one number as input and produce one number as output, and are defined for every input in their domain. Their totality guarantees that every element in the domain has an image under the function, which is important when considering compositions and properties such as being constant or the identity.

Function Composition

This is the process of applying one function to the result of another function, denoted as (g ? h)(x) = g(h(x)). In problems like this, understanding how the properties of g and h interact through composition is key to constructing examples that exhibit specific behaviors, such as yielding the identity function or a constant function.

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