Prove that if S is a subset of A then S is an empty set.

[DeadOriginal]

Homework Statement

Let S be a set such that for each set A, we have S$\subseteq$A. Show that S is an empty set.

3. Relevant equations
10.6 Proposition. For each set A, we have empty set $\subseteq$ A.

The Attempt at a Solution

Solution. Consider any set A and a set S such that S$\subseteq$A. Choose any x$\in$S. Then since S$\subseteq$A we also have x$\in$A. From proposition 10.6 we know that if x$\subseteq$empty set, then x$\subseteq$A. Now x is arbitrary. Thus S must be an empty set.

I feel like this proof is horrible and doesn't flow at all. Can someone give me some pointers?

 

[jgens]

I feel like this proof is horrible and doesn't flow at all. Can someone give me some pointers?

Clearly $\emptyset \subseteq S$ and by hypothesis $S \subseteq \emptyset$. Can you put this together?

 

[DeadOriginal]

Clearly $\emptyset \subseteq S$ and by hypothesis $S \subseteq \emptyset$. Can you put this together?

I am trying to see how this works but I don't get it.

I understand that if we can show that $\emptyset \subseteq S$ and $S \subseteq \emptyset$, then we can say that $S=\emptyset$ but I don't understand what you mean by, by hypothesis $S \subseteq \emptyset$.

 

[gopher_p]

Nevermind. Beat to the punch.

 

[jgens]

I am trying to see how this works but I don't get it.

I understand that if we can show that $\emptyset \subseteq S$ and $S \subseteq \emptyset$, then we can say that $S=\emptyset$ but I don't understand what you mean by, by hypothesis $S \subseteq \emptyset$.

In your own words: Let S be a set such that for each set A, we have S⊆A. The empty set is a set, yes?

 

[DeadOriginal]

In your own words: Let S be a set such that for each set A, we have S⊆A. The empty set is a set, yes?

Let me apologize for being a bit slow with grasping this. Ever since my proof class moved into sets I've been a little slow...

Anyhow, from the way I am thinking about it, the problem is to show that there exists a set such that it is a subset of A. Then by saying that $S\subseteq\emptyset$ we are claiming that $\emptyset$ is such a set. Is that the correct way of looking at it?

 

[jgens]

Anyhow, from the way I am thinking about it, the problem is to show that there exists a set such that it is a subset of A. Then by saying that $S\subseteq\emptyset$ we are claiming that $\emptyset$ is such a set. Is that the correct way of looking at it?

I can't make sense out of what you're saying, so could you be a little more clear?

 

[DeadOriginal]

I can't make sense out of what you're saying, so could you be a little more clear?

I can't see how saying Let S be a set such that for each set A, we have S⊆A can lead to $S\subseteq\emptyset$.

 

[jgens]

I can't see how saying Let S be a set such that for each set A, we have S⊆A can lead to $S\subseteq\emptyset$.

That is clearer, thank you. Do you accept that $\emptyset$ is a set?

 

[DeadOriginal]

Yes I do.

 

[jgens]

Yes I do.

Since $S$ is a subset of every set and the empty set is a set, this mean $S \subseteq \emptyset$ (i.e. S is a subset of the empty set). Not sure what else to say about that. If you're still confused perhaps another member could chime in and help?

 

[DeadOriginal]

Ok. Here is an attempt at a better proof.

Consider any set A and a set S such that S is a subset of A. From proposition 10.6 we know that the empty set is a subset of S. Now S is a subset of every set A so it follows that S is a subset of the empty set. Thus we have the empty set is a subset of S and S is a subset of the empty set. Therefore S is an empty set.

Does that work?

 

[jgens]

Ok. Here is an attempt at a better proof.

Consider any set A and a set S such that S is a subset of A. From proposition 10.6 we know that the empty set is a subset of S. Now S is a subset of every set A. It follows that S is a subset of the empty set. Thus we have the empty set is a subset of S and S is a subset of the empty set. Therefore S is an empty set.

Does that work?

Yeah. The only thing I would change is that I would say S is the empty set. The empty set is provably unique.

 

[DeadOriginal]

Yeah. The only thing I would change is that I would say S is the empty set. The empty set is provably unique.

Done.

Thanks so much for your help! I need to stare at this till it makes more sense to me.

 

[jgens]

Thanks so much for your help! I need to stare at this till it makes more sense to me.

Sometimes that is the only approach, haha. If you still have difficulty with the $S \subseteq \emptyset$ feel free to ask the other members. I just don't know how else to explain it :/

 

[SteveL27]

Done.

Thanks so much for your help! I need to stare at this till it makes more sense to me.

There's only one empty set. That's because a set is completely characterized by its elements. There's only one set with no elements, that's the empty set.