Final answer:
Addition and multiplication are operations well defined for the set of natural numbers, as they always result in a natural number. Subtraction and division are not always well defined, as they can result in numbers that are not natural numbers. In scientific notation, addition and subtraction require the same exponents, while multiplication and division involve manipulating both N and the exponents.
Step-by-step explanation:
The operations on numbers that are well defined in the set of natural numbers N include addition and multiplication. These operations are closed within the set of natural numbers, meaning when you add or multiply any two natural numbers, the result is also a natural number. For example, adding 2 + 3 yields 5, and multiplying 2 x 3 yields 6, both of which are natural numbers.
Subtraction and division, on the other hand, are not always well defined for natural numbers. While they can sometimes result in natural numbers, they can also lead to numbers that are not in the set of natural numbers, such as negative numbers or fractions. For instance, subtracting a larger number from a smaller (e.g., 2 - 3) or dividing by a number that does not evenly divide the other (e.g., 3 / 2) will not result in a natural number.
In the context of scientific notation, the basics are different. For addition and subtraction in scientific notation, it is important to ensure the exponents are the same before operating on the values of N. Multiplication and division in scientific notation deal with manipulating both the values of N and the exponents, but are typically not operations done within the set of natural numbers alone.