The problem asks for the diagram that represents a function from set $P$ to set $Q$. In mathematics, a function is defined as a relation where each element in the domain (set $P$) is related to exactly one element in the codomain (set $Q$).

Now, let's examine the options:

• Option a: In this diagram, some elements in $P$ are mapped to multiple elements in $Q$. This is not a function because one element in $P$ cannot be mapped to more than one element in $Q$.

• Option b: Here, all elements in $P$ are mapped to a single element in $Q$. While this is valid as a function, it's a constant function because all elements in $P$ map to the same element in $Q$, which could be considered a valid function.

• Option c: One element in $P$ is mapped to multiple elements in $Q$, so this is not a function.

• Option d: Each element in $P$ is mapped to exactly one element in $Q$, making this a valid function.

• Option e: This diagram shows one element in $P$ being mapped to multiple elements in $Q$, so it is not a function.

The correct answer is Option d, where each element in set $P$ is mapped to exactly one element in set $Q$.

Do you have any questions or need further details?

Here are five related questions you might consider:

1. What are the conditions for a relation to be a function?
2. Can a function map multiple elements of $P$ to the same element in $Q$?
3. How do you define an injective function from $P$ to $Q$?
4. What is the difference between a surjective and an injective function?
5. How do you identify a constant function in a mapping diagram?

Tip: Remember, for any relation to be a function, each element of the domain must be associated with exactly one element in the codomain.