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Math Help - Set builder notation

  1. #1
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    Set builder notation

    In the beginning part of my book. There was a solution which says:

    Given a set of negative integers, S
    , where S \subset \mathbb{Z}^-, we can build a set T of positive numbers by a set builder notation such as

    T=\{x \in Z: -x \in S\}.



    Question:


    Since
    x is only a dummy variable, I am wondering whether it's possible at all to built the set T with S \subset \mathbb{N}?

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  2. #2
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    Quote Originally Posted by novice View Post
    In the beginning part of my book. There was a solution which says:

    Given a set of negative integers, S
    , where S \subset \mathbb{Z}^-, we can build a set T of positive numbers by a set builder notation such as

    T=\{x \in Z: -x \in S\}.



    Question:


    Since
    x is only a dummy variable, I am wondering whether it's possible at all to built the set T with S \subset \mathbb{N}?

    You can, but then T would be a subset of the negative integers.
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  3. #3
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    Quote Originally Posted by gmatt View Post
    You can, but then T would be a subset of the negative integers.
    Did you notice that S is a subset of negative integers?
    That means what about T?
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    Quote Originally Posted by Plato View Post
    Did you notice that S is a subset of negative integers?
    That means what about T?
    I found that set notations can be very tricky. I have many dumb questions. Apparently, they also teach me the most. It made me think.

    I am thinking: Since elements of S are negative, where S \subset \mathbb{Z}^-, and that -x\in S, then x ought to be positive, i.e. x>0, so T=\{x\in Z: x>0\}.

    I contemplated about the question you asked--and of course, the poster immediately before you too helped me think. So, I wrote on a piece of paper

    If S\not= \emptyset and S \subset \mathbb{N}, then the elements of S cannot be negative, despite T being defined as

    T=\{x\in \mathbb{Z}: -x\in S\}. Since

    -x >0, it has to be nothing else but x<0,

    so T=\{x\in \mathbb{Z}: x<0\} as gmatt said.

    I am a little too slow, but I will have another 70 years (I meant 70 more years) to learn.

    Thanks to both of you.
    Last edited by novice; May 8th 2010 at 04:39 PM. Reason: 70 more years later I will not be 140
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    Quote Originally Posted by Plato View Post
    Did you notice that S is a subset of negative integers?
    That means what about T?
    I'm not exactly sure what you mean, if S \subset \mathbb{N} then it can't be a subset of the negative integers. I assumed what the OP meant was that he was switching the meaning of S, not adding another condition onto it ( since trivially if S \subset \mathbb{N} and S \subset \mathbb{Z}^- then S = \emptyset)
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    Quote Originally Posted by novice View Post
    I found that set notations can be very tricky. I have many dumb questions. Apparently, they also teach me the most. It made me think.

    I am thinking: Since elements of S are negative, where S \subset \mathbb{Z}^-, and that -x\in S, then x ought to be positive, i.e. x>0, so T=\{x\in Z: x>0\}.

    I contemplated about the question you asked--and of course, the poster immediately before you too helped me think. So, I wrote on a piece of paper

    If S\not= \emptyset and S \subset \mathbb{N}, then the elements of S cannot be negative, despite T being defined as

    T=\{x\in \mathbb{Z}: -x\in S\}. Since

    -x >0, it has to be nothing else but x<0,

    so T=\{x\in \mathbb{Z}: x<0\} as gmatt said.

    I am a little too slow, but I will have another 70 years to learn.

    Thanks to both of you.
    Well to be clear, if S \subset \mathbb{N} then T are all the elements of S negated...

    i.e.
    if
    S = \{ s_1, s_2, ... \} \subset \mathbb{N}

    then

    T = \{ -s_1, -s_2, ... \} \subset \mathbb{Z}^-
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  7. #7
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    Quote Originally Posted by gmatt View Post
    I'm not exactly sure what you mean, if S \subset \mathbb{N} then it can't be a subset of the negative integers.
    Quote Originally Posted by novice View Post
    Given a set of negative integers, S, where S \subset \mathbb{Z}^-, we can build a set T of positive numbers by a set builder notation such as
    T=\{x \in Z: -x \in S\}
    Did you bother to read the OP?
    It clearly states that S is a set of negative integers.
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  8. #8
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    Quote Originally Posted by gmatt View Post
    I'm not exactly sure what you mean, if S \subset \mathbb{N} then it can't be a subset of the negative integers. I assumed what the OP meant was that he was switching the meaning of S, not adding another condition onto it ( since trivially if S \subset \mathbb{N} and S \subset \mathbb{Z}^- then S = \emptyset)
    Sorry, I didn't mean to confuse anyone. I had one thing and I asked myself a question by switching S\subset \mathbb{Z}^- to S\subset \mathbb{Z}^+. Experimenting with things by taking off the head and tail, and putting tail where the head was and head where the tail was and see what shape I would get. I am groping like a blind man.
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  9. #9
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    Quote Originally Posted by Plato View Post
    Did you bother to read the OP?
    It clearly states that S is a set of negative integers.
    Sorry, sir, I wasn't ignoring your question. I realized that I have confused you. I should have made it more clear that I was switching things around for experiment.

    While I was switching things around, I confused myself and then asked question, but I have learned quite a bit though.

    If there is any consolation, you do help me learn.
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  10. #10
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    Quote Originally Posted by novice View Post
    Sorry, sir, I wasn't ignoring your question. I realized that I have confused you. I should have made it more clear that I was switching things around for experiment.

    While I was switching things around, I confused myself and then asked question, but I have learned quite a bit though.

    If there is any consolation, you do help me learn.
    Yes, that is the way I interpreted things, I think Plato might not have realized that the first part of your post was a statement and the later part of it was the question.
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