Hi

I was solving some problem about the set of functions and a doubt came to me (is it correct english ?). Consider function

$\displaystyle f: A\longrightarrow B$

Consider the case where $\displaystyle A=\varnothing$ and $\displaystyle B\neq \varnothing $

So $\displaystyle A \times B = \varnothing$

Now function is type of a relation, and any subset of $\displaystyle A \times B$ is a relation from A to B. Since $\displaystyle A\times B =\varnothing$ , and

$\displaystyle \varnothing \subseteq \varnothing$

$\displaystyle \varnothing $ is a relation from A to B. Now the definition of a function is

$\displaystyle \forall a \in A \exists ! b\in B ((a,b) \in f)$

which can be written as an implication

$\displaystyle \forall a[(a\in A)\Rightarrow \exists ! b\in B ((a,b) \in f)]$

since $\displaystyle A=\varnothing$ , the antecedent will be FALSE always , so the

implication will be TRUE always, so the condition for the function is satisfied and

we can say the the relation $\displaystyle \varnothing$ is a function from A to B.

$\displaystyle \blacksquare$

is it correct reasoning ? $\displaystyle \smile$