Explain the words relation and function. Include in your explanation how the meanings of the two words differ. Give a real-word example of a relation that is not a function. Is it possible to give an example of a function that is not a relation? If so, give one. If not, explain why it is not possible.

The words "relation" and "function" are concepts in mathematics that describe the relationship between two sets of data. While they are related, they have distinct meanings.

A relation is a set of ordered pairs where there is a connection or association between elements from the first set (called the domain) and elements from the second set (called the range). In other words, a relation shows how each input from the domain is related to one or more outputs in the range. The key feature of a relation is that it allows for multiple outputs for a single input.

For example, consider a relation that represents the relationship between people and their favorite colors. The domain may consist of people's names, and the range may consist of different colors. The relation could be {(John, Blue), (John, Green), (Sara, Red)}. Here, John has a relation to both Blue and Green, while Sara is related to Red. This relation allows for multiple outputs (colors) for the same input (John).

On the other hand, a function is a specific type of relation where each input from the domain has exactly one corresponding output in the range. In other words, a function does not allow for multiple outputs for the same input. Each input in the domain maps to a unique output in the range.

An example of a relation that is not a function could be a relation between a student's name and their test scores. If there are multiple test scores for a particular student, the relation would include multiple ordered pairs with the same input (student's name) but different outputs (test scores). This violates the requirement of a function, which states that each input should have exactly one output.

Regarding your question about whether a function can be a relation, the answer is yes. In fact, every function is a relation. This is because a function is a specific type of relation that satisfies the condition of having exactly one output for each input. Therefore, a function is a special case of a relation where there is a one-to-one correspondence between the inputs and outputs.