# Thread: Difference between "contained" and "element of"

1. ## Difference between "contained" and "element of"

I've recently begun studying some basic theory and today my professor pointed out that "The empty set is necessarily contained in, but NOT necessarily an element of every set."

To me, this seems like such a subtle point and I asked him to clarify the difference but even after various explanations it still doesn't click for me. So what exactly is it that distinguishes the two statements "contained in" and "an element of"?

Thanks!

2. Originally Posted by jrkevinfang1234
I've recently begun studying some basic theory and today my professor pointed out that "The empty set is necessarily contained in, but NOT necessarily an element of every set."

To me, this seems like such a subtle point and I asked him to clarify the difference but even after various explanations it still doesn't click for me. So what exactly is it that distinguishes the two statements "contained in" and "an element of"?

Thanks!
Your notation is off, but I know what you mean. One applies to elements of a set and the other applies to subsets. This can get confusing when we talk about classes (sets of sets). Your example though consider $\displaystyle \varnothing$ itself. Now to say $\displaystyle A\subseteq B$ is logically equivalent to saying whenever you have $\displaystyle x\in A$ then $\displaystyle x\in B$. Clearly then we have that $\displaystyle \varnothing\subseteq A$ for every set $\displaystyle A$ for if that were untrue I would have to find some $\displaystyle x\in\varnothing$ such that $\displaystyle x\notin A$. But, there is no $\displaystyle x\in A$, so the conclusion follows. Thus, clearly $\displaystyle \varnothing\subseteq\varnothing$ but $\displaystyle \varnothing\notin \varnothing$ since the statement $\displaystyle x\in \varnothing$ is always false.

So to get right down to it if I say $\displaystyle A\subseteq B$ I am saying that $\displaystyle A$ is a set that contains elements of $\displaystyle B$ and to say that $\displaystyle b\in B$ means that $\displaystyle b$ is actually in $\displaystyle B$.

3. Originally Posted by jrkevinfang1234
I've recently begun studying some basic theory and today my professor pointed out that "The empty set is necessarily contained in, but NOT necessarily an element of every set."
To me, this seems like such a subtle point and I asked him to clarify the difference but even after various explanations it still doesn't click for me. So what exactly is it that distinguishes the two statements "contained in" and "an element of"?
I would completely disagree with you instructor.
But may I ask, is the instruction in English?
If yes, then I think that most English speaking mathematicians would agree with me.
It is true that then empty set is a subset of every set.
It is true that the empty set may be contained in a set but not necessarily.
If fact, the phrase “x is contained in Y” is equivalent to “x is an element of Y”.

4. Originally Posted by Plato
I would completely disagree with you instructor.
But may I ask, is the instruction in English?
If yes, then I think that most English speaking mathematicians would agree with me.
It is true that then empty set is a subset of every set.
It is true that the empty set may be contained in a set but not necessarily.
If fact, the phrase “x is contained in Y” is equivalent to “x is an element of Y”.
Yes, the instruction is in English (and in fact, my prof's words are quoted nearly verbatim in my opening post).

I believe Drexel pretty much nailed the problem down though; my confusion was over the fact that the empty set is not an element of anything at all. In other words, the issue boils down to "what is the difference between 2 and {2}?"

At the same time though Plato, I wouldn't say you were wrong. I am merely still trying to separate "subsets" from "elements" in my mind.

Thanks guys

5. Originally Posted by jrkevinfang1234
my confusion was over the fact that the empty set is not an element of anything at all.
Really? What about $\displaystyle \left\{\varnothing\right\}$? What I said is that given any set $\displaystyle A$ it is necessarily true that $\displaystyle \varnothing\subseteq A$ but not necessarily true that $\displaystyle \varnothing\in A$. That being said, given any set $\displaystyle A$ the set $\displaystyle \wp\left(A\right)$ must contain $\displaystyle \varnothing$.

6. Originally Posted by Drexel28
Thus, clearly $\displaystyle \varnothing\subseteq\varnothing$ but $\displaystyle \varnothing\notin \varnothing$
So is there a difference between $\displaystyle \varnothing$ and {$\displaystyle \varnothing$} ?

7. Originally Posted by jrkevinfang1234
So is there a difference between $\displaystyle \varnothing$ and {$\displaystyle \varnothing$} ?
Of course, for many reasons. Most apropos is this: is the statement $\displaystyle x\in\varnothing$ ever true? What about $\displaystyle x\in\left\{\varnothing\right\}$?

Think about a set as being a bag contains things. Then $\displaystyle \varnothing$ is an empty bag and $\displaystyle \left\{\varnothing\right\}$ is a bag containing an empty bag.

8. Originally Posted by jrkevinfang1234
Yes, the instruction is in English (and in fact, my prof's words are quoted nearly verbatim in my opening post).

I believe Drexel pretty much nailed the problem down though; my confusion was over the fact that the empty set is not an element of anything at all. In other words, the issue boils down to "what is the difference between 2 and {2}?"
At the same time though Plato, I wouldn't say you were wrong. I am merely still trying to separate "subsets" from "elements" in my mind.
But that is not correct either: Ex. $\displaystyle \emptyset \in \left\{ {\emptyset ,2} \right\}\,\& \,\emptyset \subseteq \left\{ {\emptyset ,2} \right\}$

But also you must understand why $\displaystyle \emptyset \notin \left\{ {\{ \emptyset \} ,2} \right\}\text{ but }\emptyset \subseteq \left\{ {\{ \emptyset \} ,2} \right\}$

9. When I was introduced to $\displaystyle \emptyset$ as a subset of every set, I took it and swallowed it whole including the hook, but when I was introduced to basic probability it began to make sense.

For example:

Set A = {rain, shine, neither}, which can be written as A = {r,s, $\displaystyle \emptyset$ } or A = {1,2, $\displaystyle \emptyset$ }, 1 denotes rain and 2, shine. Rain and shine are elements of the set, but neither is neither rain nor shine. "Neither is neither" or None. Yah?

10. Originally Posted by novice
When I was introduced to $\displaystyle \emptyset$ as a subset of every set, I took it and swallowed it whole including the hook, but when I was introduced to basic probability it began to make sense.
For example: Set A = {rain, shine, neither}, which can be written as A = {r,s, $\displaystyle \emptyset$ } or A = {1,2, $\displaystyle \emptyset$ }, 1 denotes rain and 2, shine. Rain and shine are elements of the set, but neither is neither rain nor shine. "Neither is neither" or None. Yah?
novice, that is a well chosen screen name.
You need to do some serious study in foundations of set theory.

11. A better (although still entirely loose and heuristic) (Plato forbid) justification of why $\displaystyle \varnothing\subseteq A$ for any set $\displaystyle A$ is to continue my bag metaphor. If we think of a subset of $\displaystyle A$ as being any possible bag which I could make by removing elements from my bag $\displaystyle A$. Clearly then one of these scenarios is when I remove all things from $\displaystyle A$ leaving me with any empty bag...or an empty set.

12. Originally Posted by Plato
novice, that is a well chosen screen name.
Well, Plato, yes, I honestly like my screen name. I am still a freshmen and have not even chosen math as a major. Yes, Plato, that shows the more that you ought to help your brother in need---hee-hee-hee, your brother Novice.

Originally Posted by Plato
novice, that is a well chosen screen name.
You need to do some serious study in foundations of set theory.
Let's get to serious business. How shall begin? I just don't know how the foundations must be laid. Interesting that you say so because I was not aware that I was standing on sinking sand.

Question: Isn't {$\displaystyle 1,2,\emptyset$}=.{$\displaystyle 1,2,$,{}}. An empty space set aside in the set is empty space that is the subset of the set but not an element. Yah? Plato, Yah?

13. Originally Posted by Drexel28
A better (although still entirely loose and heuristic) (Plato forbid) justification of why $\displaystyle \varnothing\subseteq A$ for any set $\displaystyle A$ is to continue my bag metaphor. If we think of a subset of $\displaystyle A$ as being any possible bag which I could make by removing elements from my bag $\displaystyle A$. Clearly then one of these scenarios is when I remove all things from $\displaystyle A$ leaving me with any empty bag...or an empty set.
Drexel, you need to get plato to agree; otherwise, I will forever be in a quandary.

14. Originally Posted by novice
Drexel, you need to get plato to agree; otherwise, I will forever be in a quandary.
I think Plato would agree with my analysis of the situation (in the capacity as to why it make intuitive sense that every set has the empty set as a subset and why $\displaystyle \varnothing\ne\left\{\varnothing\right\}$). But, I will leave that to him to decide.

15. Originally Posted by Drexel28
I think Plato would agree with my analysis of the situation (in the capacity as to why it make intuitive sense that every set has the empty set as a subset and why $\displaystyle \varnothing\ne\left\{\varnothing\right\}$). But, I will leave that to him to decide.
I agree.
$\displaystyle \emptyset \ne \left\{ \emptyset \right\}\,,\,\emptyset = \left\{ \quad \right\} \ne \left\{ {\left\{ \quad \right\}} \right\}$

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