To address the problem, let's break down and analyze the sets [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
### Set [tex]\( x \)[/tex]
The set [tex]\( x \)[/tex] is defined as:
[tex]\[ x = \{4^n - 3n - 1 : n \in \mathbb{N}\} \][/tex]
This means that for every natural number [tex]\( n \)[/tex], we apply the function [tex]\( 4^n - 3n - 1 \)[/tex] to generate the elements of set [tex]\( x \)[/tex].
### Set [tex]\( y \)[/tex]
The set [tex]\( y \)[/tex] is defined as:
[tex]\[ y = \{g(n-1) : n \in \mathbb{N}\} \][/tex]
Here, [tex]\( g \)[/tex] is some function and [tex]\( y \)[/tex] consists of the outputs of [tex]\( g \)[/tex] when applied to [tex]\( n-1 \)[/tex] for each natural number [tex]\( n \)[/tex].
### Task
The problem is to determine the nature of the sets [tex]\( x \)[/tex] and [tex]\( y \)[/tex] without explicit information about the function [tex]\( g \)[/tex].
Given that essential information about [tex]\( g(n-1) \)[/tex] is missing, we can't explicitly determine the elements or nature of set [tex]\( y \)[/tex].
Since we are unable to fully determine the elements of [tex]\( y \)[/tex] due to insufficient information about the function [tex]\( g \)[/tex], and given the result we arrive at that is effectively [tex]\( None \)[/tex]:
- It indicates that despite the given definitions, a definitive comparison cannot be made regarding [tex]\( x \)[/tex] and [tex]\( y \)[/tex] due to the unspecified nature of [tex]\( g \)[/tex].
### Conclusion
Considering the lack of sufficient information to definitively characterize [tex]\( y \)[/tex] and compare it with [tex]\( x \)[/tex], the correct response, consistent with the output and reflection on the problem constraints given the undefined nature of [tex]\( g \)[/tex], would be:
```
None
```
Thus, due to incomplete information, a definitive answer can’t be provided directly from the given sets and functions.
