1. Nagging question

Let $\displaystyle A$ and $\displaystyle B$ be nonempty finite sets. For $\displaystyle f:A \rightarrow B$ to be a function, it must satisfy this condition:

$\displaystyle x\in A \Rightarrow (x, f(x)) \in f$.

Now, if I make $\displaystyle A=\emptyset$ and $\displaystyle B=\emptyset$, the statement of implication becomes

$\displaystyle x\notin A \Rightarrow (x,f(x)) \notin f$, from which we know that the statement of implication is true, since

False $\displaystyle \Rightarrow$ False is a true statement. In this regard, I have a question:

Is $\displaystyle f:\emptyset \rightarrow \emptyset$ a function?

2. Have you forgotten that $\displaystyle x\notin\emptyset$ is a true statement?
So $\displaystyle x\in\emptyset$ is false.

3. Originally Posted by Plato
Have you forgotten that $\displaystyle x\notin\emptyset$ is a true statement?
So $\displaystyle x\in\emptyset$ is false.
In deed, I forgot about it. Oh, Sir, I had a brain fade at noon right after eating a full meal.

4. So we see:

0 is a function from 0 into 0

By the way, another cute one is this:

[EDITED OUT; CORRECTED BELOW]

5. Originally Posted by MoeBlee
So we see:

0 is a function from 0 into 0

By the way, another cute one is this:

{1} is a function

{1} = {{0}} = {{0} {0 0}} = {<0 0>} is a function
No comprehende, amigo!

6. Originally Posted by novice
No comprehende
That's because I made a mistake.

I meant to say {1} is an ordered pair and {{1}} is a function.

{{1}} = {{{0}}} = {{{0} {0 0}}} = {<0 0>} is a function

7. Originally Posted by MoeBlee
That's because I made a mistake.

I meant to say {1} is an ordered pair and {{1}} is a function.

{{1}} = {{{0}}} = {{{0}, {0, 0}}} = {<0, 0>} is a function
MoeBlee
I am curious at why you do not use commas in ordered pairs as is the standard notation.
Why do you use $\displaystyle <0 ~0>$ in place of the standard $\displaystyle (0,0)$?
You know that can confuse many who use resource.

Again, why don’t you learn to use LaTeX?

8. I find that unneeded commas are usually visual noise.

The notation

<0 0>

is quite clear.

I've corresponded with numerous professional and amateur mathematicians and have never had any problem with

<0 0>

being understood as the ordered pair with 0 as first coordinate and 0 as second coordinate.

/

9. Originally Posted by MoeBlee
That's because I made a mistake.

I meant to say {1} is an ordered pair and {{1}} is a function.

{{1}} = {{{0}}} = {{{0} {0 0}}} = {<0 0>} is a function
I must thank you for your willingness in helping me with my learning, though I have not learned enough mathematics to understand what is being imparted to me. Hopefully, the day will come when I can recognize it and appreciate it. Mucho Gracias.

10. Originally Posted by novice
I have not learned enough mathematics to understand what is being imparted to me.
It's just a kind of fun fact that derives from our particular set theoretic definition of the ordered pair operation:

The standard definition of the ordered pair operation:

<x y> = {{x} {x y}}

Also, we have a "redundancy property":

{x x} = {x}

Also, we have the von Neumann approach to natural numbers, in which 1 = {0}.

So:

{{1}} = {{{0}}} = {{{0} {0}}} = {{{0} {0 0}}} = {<0 0>}

And {<0 0>} is a function since it satisfies the set theoretic definition of a function:

f is a function iff (f is a set of ordered pairs & there are no members <x y> and <x z> in f unless y=z)

11. Originally Posted by MoeBlee
I find that unneeded commas are usually visual noise.

The notation

<0 0>

is quite clear.

I've corresponded with numerous professional and amateur mathematicians and have never had any problem with

<0 0>

being understood as the ordered pair with 0 as first coordinate and 0 as second coordinate.

/

That may well be the case. Nevertheless, your notation is not standard notation and has the potential to create confusion. I support Plato's remarks and request that you follow them when making future posts.

12. P.S. Actually, the notation

(x, y)

is more ambiguous than

<x y>.

(x, y)

could be taken as the ordered pair of x and y, or as an open real interval.

But

<x y>

is quite clearly the ordered pair of x and y.

13. Originally Posted by MoeBlee
P.S. Actually, the notation

(x, y)

is more ambiguous than

<x y>.

(x, y)

could be taken as the ordered pair of x and y, or as an open real interval.

But

<x y>

is quite clearly the ordered pair of x and y.
This thread is taking a direction beyond the question asked by the OP.