# a little bit of functions on sets

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• Oct 14th 2008, 08:27 PM
jbpellerin
a little bit of functions on sets
1. (a) For an arbitrary set Y , determine $Y^\emptyset$
(b) For an arbitrary set X, determine $\emptyset ^X$

(a) I couldn't figure it out
(b) I think it's the empty set but I can't show how
if anyone could show me how I would greatly appreciate it
• Oct 15th 2008, 03:33 PM
Jhevon
Quote:

Originally Posted by jbpellerin
1. (a) For an arbitrary set Y , determine $Y^\emptyset$
(b) For an arbitrary set X, determine $\emptyset ^X$

(a) I couldn't figure it out
(b) I think it's the empty set but I can't show how
if anyone could show me how I would greatly appreciate it

clarify what you mean. by the notation $B^A$, do you mean the set of functions from the set $A$ to the set $B$?
• Oct 15th 2008, 04:08 PM
Plato
We can find disagreement on this question.
Paul Halmos claims the following.
“(i) $Y^\emptyset$ has exactly one element, namely $\emptyset$, whether $Y$ is empty or not and (ii) if $X \ne \emptyset$, then $\emptyset ^X = \emptyset$".
• Oct 15th 2008, 04:31 PM
Jhevon
Quote:

Originally Posted by Plato
We can find disagreement on this question.
Paul Halmos claims the following.
“(i) $Y^\emptyset$ has exactly one element, namely $\emptyset$, whether $Y$ is empty or not and (ii) if $X \ne \emptyset$, then $\emptyset ^X = \emptyset$".

is my interpretation of the problem correct? in other words, is Halmos saying the set of function from the empty set to another set is empty, and thus, represented by the empty set?

it seems strange that we should require X is nonempty in part (ii), since that is just a special case of part (i) (where Y is empty).

how exactly is this claim justified? what are the other interpretations?
• Oct 15th 2008, 04:49 PM
Plato
Quote:

Originally Posted by Jhevon
how exactly is this claim justified? what are the other interpretations?

All I can say is read his book: Naive Set Theory.
I do not defend his answer.
• Oct 15th 2008, 05:13 PM
Jhevon
Quote:

Originally Posted by Plato
All I can say is read his book: Naive Set Theory.
I do not defend his answer.

what are your views?

is this one of those issues over which mathematicians are divided? one bunch accepts such a claim, and another bunch doesn't?
• Oct 15th 2008, 06:06 PM
Plato
Well I must say that my experience is different from the vast majority on mathematicians. In all my graduate training, my professors did not believe that a set could be empty. How can a point set be empty? If it is a point set then by definition it is not empty. Of course, that is a very narrow and not practical position.
For me a function is a set of ordered pairs: $f:A \mapsto B \Rightarrow \quad f \subseteq A \times B$.
I am not sure what $\emptyset \times Y\quad \vee \quad Y \times \emptyset$ mean.

In many ways, this is a pointless question.