Final answer:
In the set of real numbers R, well-defined operations include basic arithmetic operations like addition, subtraction, multiplication, and division (except by zero), as well as operations with integer and non-integer exponents, and more complex functions like sine, cosine, logarithms, and exponential functions.
Step-by-step explanation:
The operations that are well defined in the set of real numbers R include the basic arithmetic operations such as addition, subtraction, multiplication, and division (except division by zero). The properties of real numbers ensure that these operations always result in another real number. In addition to these basic operations, operations resulting from integer powers, such as squaring or cubing a number, are also well defined. Non-integer exponents like 31.7 are computable and provide a real number result; calculators are designed to handle such calculations without difficulty.
Furthermore, more complex mathematical functions such as sine and cosine, logarithms, and exponential functions are well-defined when their inputs are dimensionless real numbers, and they yield real number outputs as well. Quadratic equations, often constructed using physical data, are expected to have real roots; when dealing with real-world applications, especially positive roots are of high interest. Two-Dimensional (x-y) Graphing techniques also leverage the properties of real numbers.
It is also important to remember that certain principles hold for operations on real numbers. For addition and subtraction, we observe the commutative and associative properties. Moreover, when we are dealing with multiplication and division, rules regarding the signs of numbers must be considered. For example, when multiplying two numbers, if both have the same sign, the result is positive; if they have opposite signs, the result is negative. The same rule applies to division.