Final answer:
The set of integers Z allows for addition, subtraction, and multiplication, which always yield another integer; however, division is not always well defined in Z. Non-integer powers, like 3¹.⁷, though mathematically valid, may result in non-integer numbers. Mathematics' rules are universal, irrelevant of time or culture, as with the example of counting goats or students.
Step-by-step explanation:
The operations that are well defined in the set of integers Z include addition, subtraction, and multiplication. These operations will always result in another integer. For instance, the sum or difference of any two integers will be an integer, following the rule that 12 + 19 = 31 or 31 - 12 = 19. Multiplication, as shown in 3´ as 3×3×3×3, is also well defined within the set of integers. However, division is not always well defined in Z because dividing two integers does not always result in another integer.
When considering non-integer powers such as 3¹.⁷, it's important to note that these are well-defined mathematical expressions but their results may not be integers. In those cases, we extend our understanding of numbers beyond the set of integers to include real and complex numbers where such operations are also well defined.
Furthermore, the rules of mathematics, including these operations, are universally valid across different times and cultures. Whether one is counting goats on a farm or students in a classroom, the same mathematical rules apply. This highlights the objective nature of mathematical operations and the importance of using the correct rules. An incorrect operation such as 12 + 19 = 32 would be an error regardless of the context.