My discrete book is defining a function, f, as a special type of relationship in which if both $\displaystyle (a,b) \in f$ and $\displaystyle (a,c) \in f$, then $\displaystyle b=c$ (and a relation is defined as a set of ordered pairs).

So, is the empty set not a function because it doesn't have any ordered pairs, or is it a function because it does not violate the definition of a function?

For each of the following relations, please answer these questions:

(1) Is it a function? If not, explain why.

(2) If yes, what are it's domain and range?

(3) Is the function one-to-one? If not, explain why.

(4) If yes, what is the inverse function?

a,b,c,d,e,... I already did

f. $\displaystyle f=\emptyset$

(1) Yes (trivially), because there are no ordered pairs in $\displaystyle f$, it does not violate the definition of function.

(2) dom $\displaystyle f$ = im $\displaystyle f$ = $\displaystyle \emptyset$

(3) Yes (trivially), since the definition of one-to-one is not violated

(4) $\displaystyle f^{-1}=\emptyset$

Above is how I wrote up my homework (but it's not due until Thu.), but I'm not confident it's the right answer.