Final answer:
The relation being described as having a domain of real numbers does not automatically determine if it's a function, reflexive, symmetric, or transitive; it depends on other characteristics. Variables X, Y, and Z are random variables, with X being all possible majors, Y being non-negative integers (class counts), and Z being non-negative real numbers (money spent on books). A negative value for Z isn't possible.
Step-by-step explanation:
The domain of a relation being the set of real numbers does not necessarily tell us if the relation is a function, reflexive, symmetric, or transitive. To determine if a relation has any of these properties, we have to look at specific characteristics:
- A relation is a function if each input has exactly one output. This is known as the vertical line test in graphical representations.
- A relation is reflexive if every element is related to itself.
- A relation is symmetric if for every pair (a, b) in the relation, the pair (b, a) is also in the relation.
- A relation is transitive if whenever (a, b) and (b, c) are in the relation, then (a, c) is also in the relation.
As for the given variables X, Y, and Z from your question:
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Variables X, Y, and Z are considered random variables because they can take any value within their domain, and their actual value is not determined until after data collection. A value of z = -7 for Z would not be possible since you cannot spend a negative amount on books. The two essential characteristics of a discrete probability distribution are that the sum of all probabilities is 1 and that each probability is between 0 and 1, inclusive.