Empty Set is Not a Subset

Contra proof to Every Set is a Subset. Because either can be true neither is true.

[wernerhartl]

This is really an example of Vacuous Truth.
From wiki article “Vacuous Truth:”

“In mathematics and logic, a vacuous truth is a conditional or universal statement that is only true because the antecedent cannot be satisfied. For example, the statement "all cell phones in the room are turned off" will be true even if there are no cell phones in the room. In this case, the statement "all cell phones in the room are turned on" would also be vacuously true, as would the conjunction of the two: "all cell phones in the room are turned on and turned off". For that reason, it is sometimes said that a statement is vacuously true only because it does not really say anything.”

[AlienwareH]

To show that E is not a subset of A, you must show the EXISTENCE of an element of E not contained in A.



Sure, any proposition about the elements of the empty set is vacuously true. However, this isn't enough. Showing the truth of the statement "If x is an element of the empty set then x is not in A" is not enough to show that E satisfies our definition of "not subset". You must show there exists some element of E not contained in A in order to truly say E is not a subset of A.

[MuffinsAPlenty]

Yes, both the statements "No member of E is a member of A" and "Every member of E is not a member of A" are vacuously true. But here's the kicker. So is the statement "Every member of E is a member of A". The difference? The last of these three statements is the definition of "E is a subset of A". The first two statements are _not_ negations of the definition of "E is a subset of A".

The two statements you show are true are irrelevant to whether or not E is a subset of A. Nevertheless, the same argument you use, when applied properly, shows E is a subset of A.

[weilam]

Let 𝐴 and 𝐵 be sets. If every element 𝑎 ∈ 𝐴 is also an element of 𝐵, then 𝐴 ⊆ 𝐵.
The contrapositive of the statement would be
If 𝐴 ⊈ 𝐵, then there exists some element 𝑥 ∈ 𝐴 such that 𝑥 ∉ 𝐵.

If 𝐴 is the empty set, there are no 𝑥's in 𝐴, so in particular there are no 𝑥s in 𝐴 that are not in 𝐵. Thus 𝐴 ⊈ 𝐵 can't be true. Furthermore, note that we haven't used any property of 𝐵 in the previous line, so this applies to every set 𝐵, including 𝐵=∅.

[siulapwa]

The problem I have with empty sets is if you break down a bigger set into smaller sets each of them has empty sets in them and then you can combine the small sets into a bigger set and it will only have one empty set in Maybe it's just a handy definition like in a vector space

[wernerhartl]

Definition: A set is a collection of objects. No objects, no set. The Empty set doesn’t exist.

[Chris-]

The empty set is a subset of every set. That's because every element in the empty set (all none of them) is also in set A. What's so hard to understand about that?

[MikeRosoftJH]

No, empty set is by definition a subset of every set. The statement "set A is a subset of set B" means: "every element of A is an element of B"; or, formally, "for every x, if x is an element of A, it is an element of B". It's indeed the case that, for example, empty set is a subset of the set {1, 2, 3}. Every element of the empty set (all zero of them) is an element of the set {1, 2, 3}. (For any x it is the case that if x is in the empty set, then x is in the set {1, 2, 3}. And this is true, because "x is an element of the empty set" is not true for any x.) Yes, it is also true that for all elements of an empty set it is the case that the element is *not* in the set {1, 2, 3}. (For every x, if x is an element of the empty set, then it is not an element of the set {1, 2, 3}.) But that doesn't contradict that empty set is a subset of the set {1, 2, 3}; the subset relation is defined as the former, not as the latter. (The opposite of "for every x, P(x) is true" is "there exists x, for which P(x) is false". So if you disagree that empty set is a subset of the set {1, 2, 3}, then find me some element for which the following is true: it is an element of the empty set, and it is not an element of {1, 2, 3}. Does 0 have this property? Does 1 have this property? Does 2, 3, or 4?) And there's nothing ambiguous about this.

I see that you are unable to wrap your head around the fact that - informally speaking - the statements "I have won on all lottery tickets that I have bought today" and "I have won on no lottery tickets that I have bought today" are both true (given that I haven't bought any tickets); so is both "if 2+2 equals 5, then I am a Chinese hatter" and "if 2+2 equals 5, then I am not a Chinese hatter". You may not like that, but that's how mathematical logic is defined. Just because you don't understand, or disagree with, the rules of mathematical logic, it doesn't mean that mathematicians will stop using it.

[Leontor]

Funny thing is ... For a statement to be false, you must show a counterexample. If it's true, though, you don't need to show an example.

And thats why we consider the empty set to be a subset of every set. Because there's no counterexample. That is, you cant pick an element in E thats not in A.

[colinjava]

Again this is false.

Using the definition "E is a subset of A if every element of E is an element of A"
So when E is the empty set, that statement is either true or false.

If it was false, that would mean there must be elements in E that are not in A.
Since E has not elements, E does not satisfy the condition for the statement to be false.

So the statement must be true.

So the empty set is a subset of A.



It also follows naturally when you take the power set of A, P(A)
For simplicity, assume A is finite and has n elements (but it works the same if it has an infinite cardinality)

So A = {a_1, a_2, a_3, ...., a_n}

Lets create all the subsets of A.
We can either include or not include a_1, so there's 2 choices.
We can either include or not include a_2, so there's 2 choices.
We can either include or not include a_3, so there's 2 choices.
...
We can either include or not include a_n, so there's 2 choices.

Overall, there's gonna be 2^n choices. One of those being the empty set.

Get over it.

[akhilendrapratapsingh]

Oh! prophesar I like your teaching method, I am akhilendra pratap singh from india(BHARAT)

[pendaranroberts]

I watched the video and read a lot of the comments. I think that given the definition of a subset, the empty set is a subset of every set. However, the definition has this weird implication that a set E can be a subset of another set A even though no members of E are in A. We might think this shows that the definition is wrong. Maybe a set should be defined as having at least one member. I think this better captures the idea of a set in English. Of course, mathematicians can define it however they want. I don't see why we care?!?

[prestonrasmussen]

Things that are vacuously true are still true

[thetaomegatheta]

If empty set is not a subset of any given set A, then there exists an element of the empty set that is not a member of A.
This is obviously false for any choice of A.
So, yeah, you keep failing at mathematics. Please, just admit at this point that you misunderstand something, and, perhaps, read a book, or go back to school, rather than try to misinform people.

[wernerhartl]

E is a subset of A is vacuously True.
E is not a Subset of A is vacuously True. Therefore it makes no logical sense, and is also misleading, to claim either is True, since the opposite is also True.