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Math Help - Some noob analysis questions

  1. #1
    Senior Member DivideBy0's Avatar
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    Some noob analysis questions

    I just started Analysis and I have a few questions,

    1. "In a partial order on X, an element x \in X is maximal if (y \in X) \wedge (x \leq y) \Rightarrow y = x, and x is maximum if z \leq x for all z \in X"

    It seems to me that these definitions are equivalent, since if (y \in X) \wedge (x \leq y) \Rightarrow y = x it means that \nexists\ y \in X such that x < y, so x is bigger than every other element in the set.
    Also, \forall z \in X,\ z \leq x is pretty much saying x is bigger than every other element in the set, right?
    So what am I missing? :/

    2.
    "A subset Y of X such that for any x,y \in Y, either x \leq y or y \leq x is called a chain. If X itself is a chain, the partial order is a linear or total order."
    Just to make sure I'm understanding this alright:
    Say you have a set X=\lbrace\lbrace 1,2\rbrace,\lbrace 3\rbrace,4,5,6\rbrace
    Then would Y = \lbrace \lbrace 1,2\rbrace, \lbrace 3\rbrace \rbrace with partial order \subseteq and Z = \lbrace 4,5,6\rbrace with partial order \leq be "chains"?
    What are other partial orders you could have? I can only think of \leq, \geq, \subseteq, \supseteq


    3. Do partially ordered sets have to be countable?

    4. What does \mathbb{R}^\mathbb{N} mean?


    Thanks
    Last edited by DivideBy0; March 8th 2010 at 02:47 AM.
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    Quote Originally Posted by DivideBy0 View Post
    I just started Analysis and I have a few questions,

    1. "In a partial order on X, an element x \in X is maximal if (y \in X) \wedge (x \leq y) \Rightarrow y = x, and x is maximum if z \leq x for all z \in X"

    It seems to me that these definitions are equivalent, since if (y \in X) \wedge (x \leq y) \Rightarrow y = x it means that \nexists\ y \in X such that x < y, so x is bigger than every other element in the set.
    Also, \forall z \in X,\ z \leq x is pretty much saying x is bigger than every other element in the set, right?
    So what am I missing? :/


    In the definition of maximal there is no requirement that if y\in X\,\,\,then\,\,\,y\leq x , and in the definition of maximum THERE IS such a requirement.

    For example, if you look at the set of sets S:=\left\{\{1,2,3\}\,,\,\{2\}\,,\,\{1,3,4,5\}\righ  t\} and you partial order it wrt subset relation, then both \{1,2,3\}\,,\,\{1,3,4,5\}

    are maximal elements of S since there is no set (element) in S "bigger" (containing) any of them.


    Now, if you'd put S'=S\cup \{1,2,3,4,5\} then \{1,2,3,4,5\} is now the maximum element of S'.

    Pay attention to the fact that is an element is a maximum then it also is a maximal element and, in fact, the only one (maximum/maximal) there is, but there can be many maximal elements.

    Finally, wrt your last question: that an element is maximal means there is no element bigger than it, and that an element is a (in fact, as already noted, THE) maximum means it is the biggest element of the set.

    2.
    "A subset Y of X such that for any x,y \in Y, either x \leq y or y \leq x is called a chain. If X itself is a chain, the partial order is a linear or total order."
    Just to make sure I'm understanding this alright:
    Say you have a set X=\lbrace\lbrace 1,2\rbrace,\lbrace 3\rbrace,4,5,6\rbrace
    Then would Y = \lbrace \lbrace 1,2\rbrace, \lbrace 3\rbrace \rbrace with partial order \subseteq and Z = \lbrace 4,5,6\rbrace with partial order \leq be "chains"?


    Y is not a chain because neither \{1,2\}\subset \{3\} nor \{3\}\subset \{1,2\} . Z is a chain because it contains one unique element so it vacuously fulfills the condition.



    What are other partial orders you could have? I can only think of \leq, \geq, \subseteq, \supseteq

    There are infinitely many different partial orders on infinitely many different sets, but perhaps it's too soon to even give you examples. First grab the basics.


    3. Do partially ordered sets have to be countable?

    No

    4. What does \mathbb{R}^\mathbb{N} mean?

    Per definition it is the set of all the functions from the natural numbers to the real ones. You can think of it as the set of all the real sequences...

    Tonio


    Thanks
    .
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  3. #3
    Senior Member DivideBy0's Avatar
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    Thanks so much tonio, I think I understand it now - just 2 more questions, if you have a partial order with respect to \leq, then maximum = maximal, right?

    Also, say you have a chain Y = [0,1) \subset X, then that doesn't have a maximal element, but according to Zorn's lemma it should? How does that work?
    Last edited by DivideBy0; March 8th 2010 at 04:08 AM.
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    Excellent, Tonio.

    However, in Problem 2 he asked about Z= {4, 5, 6} with \le as order- he was thinking of them as numbers, not as a single set.

    Hurry and get those new reading glasses!

    Divideby0, \le for numbers is a linear order so, yes, "maximal" and "maximum" are the same- in fact, you would not normally use the word "maximal" in a linear order.

    Zorn's lemma says that if every chain in a partial order has a maximum, then the set contains maximal elements. It does NOT say every chain has a maximum!
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    Quote Originally Posted by DivideBy0 View Post
    Thanks so much tonio, I think I understand it now - just 2 more questions, if you have a partial order with respect to \leq, then maximum = maximal, right?


    It seems like you're thinking here of a very particular, specific example: the usual partial order on the real numbers, right? In this case we must be careful with derfinitions and NAMES, which can vary between differnet authors: most people, I believe, would call in this case SUPREMUM to what you call maximal, and maximum to maximum. This can be a little confusing, but the difference is the supremum always exists in the reals if the set is bounded, whereas the maximum is a supremum belonging to the given set


    Also, say you have a chain Y = [0,1) \subset X, then that doesn't have a maximal element, but according to Zorn's lemma it should? How does that work?
    What has Zorn's Lemma to do here? The set (closed-open or clopen interval) [0,1) in the reals is bounded and thus has a supremum and an infimum. The supremum is 1\notin [0,1) and is thus not a maximum, whereas the infimum is 0\in [0,1) and is thus also the minimum.

    Tonio
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