- #1

Gale

- 676

- 2

## Homework Statement

Using the precise denition of a function and a little logic, show that, for every set [itex]Y[/itex] , there is exactly one function [itex]f [/itex] from [itex] \emptyset [/itex] to [itex]Y[/itex] . When is [itex]f[/itex] injective? Surjective?

Let [itex] X[/itex] be a set. Show that there are either no functions from [itex]X[/itex] to [itex] \emptyset [/itex]

or exactly one such function, depending on whether [itex] X \neq \emptyset [/itex] or [itex] X= \emptyset [/itex]

## Homework Equations

## The Attempt at a Solution

So far all I can think of is that the definition of a function I think is: [tex] f:X \rightarrow Y, f=\{(x,f(x)), such\ that\ \forall x \in X, \ \exists ! y\in Y : f(x)=y \}[/tex]

Or something like that. I'm having a hard time trying to write a proper definition of a function, especially because I'm fairly sure my teacher wants something to do with ordered pairs, (as that's how functions are graphed and he mentioned in class that functions should be defined by their graphs...)

Anyway, regarding the null set: All I can think is that the null set is also a subset of Y, and therefore the one function that maps from the null set, is the one that also maps to the null set? And that the function is surjective and injective only when [itex] Y= \emptyset [/itex]. But then I'm very confused about how you can map a non-elements to non-elements, so perhaps I'm wrong, in which case, I have no idea where to go with this problem.

Thanks for any help!