Which statement correctly describes a function on a finite set?

• A. Each element of the domain is mapped to exactly one element of the codomain.
• B. Each element of the codomain has a preimage.
• C. The mapping must be one-to-one.
• D. There are no restrictions on mapping.

Answer: (A)

The key idea is that a function assigns exactly one output to every input. For a finite domain, this means each element in the domain must map to one and only one element in the codomain. That single-output rule is what makes the relation a function and keeps it deterministic.

This correctly captures the concept because a function can still send different inputs to the same output (many inputs can share one result), which is allowed. It just can’t assign two different outputs to the same input, or assign no output at all.

For example, with domain {1, 2, 3} and codomain {a, b, c}, a valid function could map 1 to a, 2 to a, and 3 to b. That’s fine, even though two inputs share the same output. If a mapping tried to give both a and b as outputs for input 1, that would violate the one-output rule and wouldn’t be a function. The idea that every codomain element must have a preimage, or that mappings must be one-to-one, or that there are no restrictions at all, are not required by the definition.