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Basic set theory question

  1. Jan 14, 2012 #1
    I'm attempting to teach myself topology from a textbook. I'm on the first chapter and came into some trouble with some of the set theory.

    Here is what the textbook says.

    We make a distinction between the object a, which is an elemant of a set A, and the one-element set {a}, which is a subset of A. To illustrate if A is the set {a,b,c}, then the following statements are all correct.

    •a is an element of A
    •{a} is a subset of A
    •{a} is an element of P(A) where P(A) i the power set of A meaning that P(A) is the set of all subsets of A.

    However according to the textbook the following statements are not true
    •{a} is a member of A
    •a is a subset of A

    If the set {a}, simply contains a what is the difference between saying a is an "element" of A and a is a "subset" of A? If an object is an element of some set isn't it also a subset of that set? I also am having trouble understanding the idea of a power set. If P(A) is the set of all subsets then doesn't P(A)=A?
  2. jcsd
  3. Jan 14, 2012 #2


    Staff: Mentor

    the textbook is right

    {a} is a subset of A and {a} is a member of the PowerSet(A) since the power set contains all subsets of A including ∅ and A itself.

    {a} is NOT a member of A.

    a is an element of A and a is NOT a member of P(A) as P(A) contains only subsets of A and not any of its elements.

    remember a ≠ {a} this is a crucial distinction.
  4. Jan 14, 2012 #3
    I still don't understand how they are different though. If the only element of {a} is a, then how does a≠{a}?

    You also didn't answer my other question. Or maybe you did but I didn't understand it.

    The textbook seems to imply that just because a is an element of A doesn't mean a is a subset of A. Could you explain this?
  5. Jan 14, 2012 #4
    Also, aren't all the elements of A also subsets of A? For example of A={1,2,3}, {1} is a subset of A, right? And wouldn't {1} be one of the elements of P(A)? ugh i'm confusing myself.
  6. Jan 14, 2012 #5


    Staff: Mentor

    You can construct sets where the elements are also subset of the set.

    consider a set N = { x, {x}, {x, {x} } ... } this is how they sometimes represent natural numbers where x is ∅ the empty set.

    sets with in sets within sets. x=0 and {x}=1 and {x,{x}} = 2 ... (see wikipedia: set-theoretic numbers)

    but in general the power set contains all possible subsets of A and while members of A could be subsets of A that isn't always true. I mean we could make a set A where some or all of the elements happen to also be subsets of A thats not true in general.
  7. Jan 14, 2012 #6


    Staff: Mentor

    okay so start with a={1,2,3}: yes {1} is a subset of A and it is a member of the P(A) because by definition the P(A) contains all subsets of A including A itself and ∅ the empty set.

    But what you said earlier is that 1 is an element of A but 1 is not an element of P(A) because 1 is not a set.

    what the book is saying when it says they aren't TRUE is that they aren't ALWAYS true and MATH really likes to have statements that are ALWAYS true.
  8. Jan 14, 2012 #7
    [Please dont think im trying to patronise you here. I still use this kind of explanation to explain university level maths to myself.]

    Replace the word 'set' with the word 'bag'.

    Now suppose you have 2 bags, one has an apple inside, one has a banana. So you have {a} and {b}.

    Now try two experiments:

    1) Tip the apple from one bag into the other bag. Now you have a bag with an apple and a banana in it, and an empty bag.

    So you have {a, b} and {}.

    2) Put one of your bags (which contains an apple) inside the other bag. Now you have one bag which contains a banana and which also contains a bag. And in that second bag there is an apple

    So you now have {{a}, b}.

    It should be clear from the bag analogy that {a, b} ≠ {{a}, b}.

    If you understand this, try it with 2 bags and one apple.

    Basically, what i'm trying to say is that a set, in the most naive sense, is an object which contains other mathematical objects. Thus a set can contain another set because that second set is a mathematical object. (You cant just pretend the bag isn't there)

    An element of a set A is an object which is contained in A.

    A subset of a set A is a collection (a set (bag) in it's own right) of elements of A.
  9. Jan 14, 2012 #8
    In order to understand power sets you need to understand what a subset is.

    I have an exercise for you that may help:

    Consider the set {a, b, c, d}.

    List all the subsets of {a, b , c, d} you can think of.

    [A little note. A book on topology will assume prior knowledge of set theory so the introductory chapter on sets will be quite brief. My suggestion to you is that you first find a book devoted to set theory for a more detailed introduction.]
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