Thread: 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



Consider the case where and

So

Now function is type of a relation, and any subset of is a relation from A to B. Since , and



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



which can be written as an implication



since , 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 is a function from A to B.


is it correct reasoning ?

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.

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....