# Math Help - empty set and functions

1. ## empty set and functions

Hi

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

$f: A\longrightarrow B$

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

So $A \times B = \varnothing$

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

$\varnothing \subseteq \varnothing$

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

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

which can be written as an implication

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

since $A=\varnothing$ , 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 $\varnothing$ is a function from A to B.
$\blacksquare$

is it correct reasoning ? $\smile$

2. ## Re: empty set and functions

is it correct reasoning ?
You are absolutely right.

To go a little further, consider the following definition. A set A is called initial if for every set B, there is one and only one function from A to B. Then the empty set is the unique initial set.

3. ## Re: empty set and functions

thanks makarov, I have finished first 4 chapters of Daniel Velleman's "How to prove it" and going to the fifth chapter , Functions. Since I am familiar with all the logical
machinery , it was easy to draw the conclusion I drew. If somebody has only calculus background , then it will be difficult for that person to see why there is a function from A to B , when A is an empty set......... By the way , I just downloaded a beautiful book on set theory , "The Joy of sets:Fundamentals of contemporary set theory" by Keith Devlin and since I have already studied basic logic , its really joy to read it....

$\bigstar$