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Math Help - Difference between "contained" and "element of"

  1. #1
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    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!
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by jrkevinfang1234 View Post
    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 \varnothing itself. Now to say A\subseteq B is logically equivalent to saying whenever you have x\in A then x\in B. Clearly then we have that \varnothing\subseteq A for every set A for if that were untrue I would have to find some x\in\varnothing such that x\notin A. But, there is no x\in A, so the conclusion follows. Thus, clearly \varnothing\subseteq\varnothing but \varnothing\notin \varnothing since the statement x\in \varnothing is always false.



    So to get right down to it if I say A\subseteq B I am saying that A is a set that contains elements of B and to say that b\in B means that b is actually in B.
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    Quote Originally Posted by jrkevinfang1234 View Post
    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.
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    Quote Originally Posted by Plato View Post
    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
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by jrkevinfang1234 View Post
    my confusion was over the fact that the empty set is not an element of anything at all.
    Really? What about \left\{\varnothing\right\}? What I said is that given any set A it is necessarily true that \varnothing\subseteq A but not necessarily true that \varnothing\in A. That being said, given any set A the set \wp\left(A\right) must contain \varnothing.
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    Quote Originally Posted by Drexel28 View Post
    Thus, clearly \varnothing\subseteq\varnothing but \varnothing\notin \varnothing
    So is there a difference between \varnothing and { \varnothing} ?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by jrkevinfang1234 View Post
    So is there a difference between \varnothing and { \varnothing} ?
    Of course, for many reasons. Most apropos is this: is the statement x\in\varnothing ever true? What about x\in\left\{\varnothing\right\}?

    Think about a set as being a bag contains things. Then \varnothing is an empty bag and \left\{\varnothing\right\} is a bag containing an empty bag.
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    Quote Originally Posted by jrkevinfang1234 View Post
    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. \emptyset  \in \left\{ {\emptyset ,2} \right\}\,\& \,\emptyset  \subseteq \left\{ {\emptyset ,2} \right\}

    But also you must understand why \emptyset  \notin \left\{ {\{ \emptyset \} ,2} \right\}\text{ but }\emptyset  \subseteq \left\{ {\{ \emptyset \} ,2} \right\}
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    When I was introduced to \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, \emptyset } or A = {1,2, \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?
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    Quote Originally Posted by novice View Post
    When I was introduced to \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, \emptyset } or A = {1,2, \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.
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    MHF Contributor Drexel28's Avatar
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    A better (although still entirely loose and heuristic) (Plato forbid) justification of why \varnothing\subseteq A for any set A is to continue my bag metaphor. If we think of a subset of A as being any possible bag which I could make by removing elements from my bag A. Clearly then one of these scenarios is when I remove all things from A leaving me with any empty bag...or an empty set.
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    Quote Originally Posted by Plato View Post
    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.

    Quote Originally Posted by Plato View Post
    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 { 1,2,\emptyset}=.{ 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?
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    Quote Originally Posted by Drexel28 View Post
    A better (although still entirely loose and heuristic) (Plato forbid) justification of why \varnothing\subseteq A for any set A is to continue my bag metaphor. If we think of a subset of A as being any possible bag which I could make by removing elements from my bag A. Clearly then one of these scenarios is when I remove all things from 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.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by novice View Post
    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 \varnothing\ne\left\{\varnothing\right\}). But, I will leave that to him to decide.
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    Quote Originally Posted by Drexel28 View Post
    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 \varnothing\ne\left\{\varnothing\right\}). But, I will leave that to him to decide.
    I agree.
    \emptyset  \ne \left\{ \emptyset  \right\}\,,\,\emptyset  = \left\{ \quad  \right\} \ne \left\{ {\left\{ \quad  \right\}} \right\}
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