Explaining technicalities involving the empty set

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In summary, the problem statement in 18a does not allow for A to be empty, but the problem statement in 17a does. This is due to the fact that in 17a, if A is empty, then all the hypotheses can be satisfied (the composition will be empty too, obviously), but g need not be equal to h. Am I right? If so, why isn’t this addressed in some problems, but is in others?
  • #1
Uncanny
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I’m curious why and where it’s necessary to explicitly state whether or not a set must be excluded from potentially being the empty set.
For instance, I attached two problems in the the thumbnail below. I’m curious why A cannot be the empty set in 18b, but A is not excluded from being the empty set in 17a.
In 17a, if A is empty, then all the hypothesis can be satisfied (the composition will be empty too, obviously), but g need not be equal to h. Am I right? If so, why isn’t this addressed in some problems, but is in others?
 

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  • #2
The answer is to do with logic, not set theory. In 18b, if A is empty, the 'for all ... if ... then ...' is what logicians call a vacuous truth, ie it says nothing useful, just as the vacuous truth 'If 1=2 then Lady Gaga is the pope' says nothing useful. Why is it vacuous? Because if A is empty there will be no functions g or h from A to B, so saying something is true for all such functions is saying nothing.

The same problem does not arise in 17a because there is no 'for all ...' statement.

Reading the linked article on vacuous truth should help make this clearer.
 
  • #3
I understand what you're saying regarding vacuous truth, but isn't a function with domain being the empty set technically a function (the empty set)? If so, then if A is the empty set in problem 18a, wouldn't the corresponding statement similarly be "saying nothing?" Why isn't the restriction on A not included here?
 
  • #4
Uncanny said:
I understand what you're saying regarding vacuous truth, but isn't a function with domain being the empty set technically a function (the empty set)? If so, then if A is the empty set in problem 18a, wouldn't the corresponding statement similarly be "saying nothing?" Why isn't the restriction on A not included here?
If you allow ##A## to be the empty set in 18b, then you can find a counterexample to the stated result. You should try this.

For problem 17, if you allow ##A## to be the empty set, then you cannot construct a counterexample.

In other words, the results in 17 and 18a holds for empty ##A## but the result in 18b fails for empty ##A##.
 
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  • #5
Yes, I think I noticed that for 18a. So is this a mistake in the problem statement (not restricting A to non-empty sets)?
 
  • #6
Uncanny said:
Yes, I think I noticed that for 18a. So is this a mistake in the problem statement (not restricting A to non-empty sets)?
Sorry, I got the numbers confused. I've edited now. The problems are right as they are.
 
  • #7
Alright, gotcha. That makes sense now- so that’s the more practical reason the restriction is necessary: when, without it, the intended result(s) fails to hold?
 
  • #8
17b) also requires A to be non-empty though, no? Otherwise, the definition of surjection can’t be satisfied.
 
  • #9
Uncanny said:
17b) also requires A to be non-empty though, no? Otherwise, the definition of surjection can’t be satisfied.
No. Let's analyse these scenarios with the assumtpion that ##A## is the empty set. This implies that ##f## is the empty function (if that's the right term).

17a) If ##f: A \rightarrow B## is onto, then ##B## must also be the empty set and ##g, h## must both be the empty function. The final condition and the result are trivial.

17b) In this case, ##f## and both compositions ##f \circ g## and ##f \circ h## are the empty function.

If ##B## is the empty set, then the condition is met trivially and the result holds. Since ##g = h## automatically and ##f## is onto.

If ##B## is not the empty set, then the premise cannot hold. I.e. ##f \circ g = f \circ h## for all functions ##g, h##, even if ##g \ne h##. There is no case to test for ##f## being onto. In other words, if ##A## is the empty set then you never have the case where:

##f \circ g = f \circ h \ \Rightarrow g = h##

So, you never have a case where ##f## must be onto.
 
  • #10
Thanks! You hit the nail on the head with the case in 17b) in which B is non-empty, but A is. This is the case I was concerned with.

I agree with everything you’ve written, I suppose my “real” question, then, is: shouldn’t the problem statement avoid or warn against this outcome by including the additional premise that A must be non-empty (unless B is also empty)?

I understand this seems cumbersome and, perhaps, distracting to include, but the fact that A need be non-empty does arise (or is asserted) in the course of the proof, as long as the vacuous truth cases are omitted.
 
  • #11
Uncanny said:
Thanks! You hit the nail on the head with the case in 17b) in which B is non-empty, but A is. This is the case I was concerned with.

I agree with everything you’ve written, I suppose my “real” question, then, is: shouldn’t the problem statement avoid or warn against this outcome by including the additional premise that A must be non-empty (unless B is also empty)?

I understand this seems cumbersome and, perhaps, distracting to include, but the fact that A need be non-empty does arise (or is asserted) in the course of the proof, as long as the vacuous truth cases are omitted.
17b holds whether A is empty or not.

Let's take a different example. Suppose you had something like:

Let ##a, b, c## be numbers. Show that if ##a = b^2## then ...

You might be concerned about the case ##a = 0##, where ##b## must also be zero.

What you're saying is that this needs to be changed to say:

Let ##a, b, c## be numbers where ##a \ne 0## unless ##b \ne 0##. Show that if ...

This is an unnecessary condition. The relationship between ##a## and ##b## comes out of the other conditions.

The simple fact is, for 17b, unless you can construct a counterexample with ##A## as the empty set, then there is no reason to exclude it.

And, unless you have a counterexample, you really can't argue!
 
  • #12
Ah, I think I understand now. If A is empty and B is non-empty, both directions of the biconditional result in vacuous truth. If we choose A empty, but not B, then f is not surjective, making the antecedent false. Analogous argument going in the other direction, per your reasoning in post #9. Have I “gotten it?”
 
  • #13
Uncanny said:
Ah, I think I understand now. If A is empty and B is non-empty, both directions of the biconditional result in vacuous truth. If we choose A empty, but not B, then f is not surjective, making the antecedent false. Similarly, going in the other direction, per your reasoning in post #9. Have I “gotten it?”
I think so, but I'm not that knowlegeable about the formal terminology for all this!
 

1. What is the empty set?

The empty set, also known as the null set, is a mathematical concept that represents a set with no elements. It is denoted by the symbol ∅ or {}.

2. Why is the empty set important in mathematics?

The empty set is important because it serves as the basis for mathematical induction, a powerful proof technique used in many areas of mathematics. It also plays a crucial role in set theory and other branches of mathematics.

3. Can the empty set be a subset of any set?

Yes, the empty set is a subset of every set. This is because every element in the empty set is also an element of any other set, since there are no elements in the empty set.

4. How is the empty set used in computer science?

The empty set is used in computer science to represent the absence of data or the end of a data structure. It is also used in programming languages to denote an empty list or array.

5. Is the empty set the same as the set containing zero?

No, the empty set and the set containing zero are different. The empty set has no elements, while the set containing zero has one element (zero). Additionally, the empty set is a subset of every set, while the set containing zero is not necessarily a subset of every set.

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