The statements you've provided pertain to relations between variables x and y and the domain of these variables. Let's analyze each statement:
I. "For every x there is only one y": This statement suggests that each value of x corresponds to a unique value of y. In mathematical terms, it implies that the relation is a function. A function assigns a single output (y) to each input (x).
II. "For every y there is only one x": This statement is similar to the first one but framed from the perspective of y. It also suggests that the relation is a function.
III. "The domain is the set of real numbers": This statement specifies the set of possible values for the variable x. If the domain is the set of real numbers, it means x can take any real number as its value.
In summary, statements I and II suggest that the relation between x and y is a function where each input (x or y) corresponds to a unique output. Statement III specifies that the domain of x includes all real numbers. These conditions are often met in mathematical functions, where there is a one-to-one correspondence between inputs and outputs for a given domain.