0
$\begingroup$

Question. Find the set of functions from $\emptyset$ into a set $Y$

How can I find?

  • 1
    That definition is off. You don't want an *element* of $\Bbb N$, but a *subset* instead. As pointed out below it's not relevant here, unless your goal is to show that $\emptyset$ is a finite set (or you're implicitly identifying an element $k$ with the subset $\{1,2, \ldots, k\}$) – 2017-01-18

1 Answers 1

1

A function from set $\;A\;$ to set $\;B\;$ is a subset $\;f\;$ of the cartesian product $\;A\times B\;$ such that $\;(a,b),\,(a,b')\in f\implies b=b'\;$ .

So a function $\;\emptyset\to Y\;$ is a subset of $\;\emptyset\times Y=\emptyset\;$, and thus there is only one such function no matter what $\;Y\;$ is, namely: the function $\;\emptyset\;$

  • 1
    @What? I didn't understand what you're asking. What I wrote at the beginning above is **the** definition of function using set theory. Any other definition *must* comply with this one. – 2017-01-18
  • 0
    You are right. I want to find this as using the definition of finite. – 2017-01-18
  • 0
    @Kahler Why would the definition of finite be at all relevant here? – 2017-01-18
  • 1
    It might be worth mentioning that the one subset of the cartesian product really is a function in this case (since this would not always be the case if one was to switch the order). – 2017-01-18