# empty set and functions

• Sep 14th 2011, 01:24 AM
issacnewton
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

$\displaystyle f: A\longrightarrow B$

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 ? $\displaystyle \smile$ (Emo)
• Sep 14th 2011, 02:58 AM
emakarov
Re: empty set and functions
Quote:

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.
• Sep 14th 2011, 03:23 AM
issacnewton
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 (Rock)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....

$\displaystyle \bigstar$