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Thread: 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, $\displaystyle S$
    , where $\displaystyle S \subset \mathbb{Z}^-$, we can build a set $\displaystyle T$ of positive numbers by a set builder notation such as

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



    Question:


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

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

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



    Question:


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

    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 $\displaystyle S$ is a subset of negative integers?
    That means what about $\displaystyle T$?
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    Quote Originally Posted by Plato View Post
    Did you notice that $\displaystyle S$ is a subset of negative integers?
    That means what about $\displaystyle T$?
    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 $\displaystyle S$ are negative, where $\displaystyle S \subset \mathbb{Z}^-$, and that $\displaystyle -x\in S$, then $\displaystyle x$ ought to be positive, i.e. $\displaystyle x>0$, so $\displaystyle T=\{x\in Z: x>0\}$.

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

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

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

    so $\displaystyle T=\{x\in \mathbb{Z}: x<0\}$ 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 03:39 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 $\displaystyle S$ is a subset of negative integers?
    That means what about $\displaystyle T$?
    I'm not exactly sure what you mean, if $\displaystyle S \subset \mathbb{N}$ 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 $\displaystyle S$, not adding another condition onto it ( since trivially if $\displaystyle S \subset \mathbb{N}$ and $\displaystyle S \subset \mathbb{Z}^-$ then $\displaystyle S = \emptyset$)
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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 $\displaystyle S$ are negative, where $\displaystyle S \subset \mathbb{Z}^-$, and that $\displaystyle -x\in S$, then $\displaystyle x$ ought to be positive, i.e. $\displaystyle x>0$, so $\displaystyle T=\{x\in Z: x>0\}$.

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

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

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

    so $\displaystyle T=\{x\in \mathbb{Z}: x<0\}$ 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 $\displaystyle S \subset \mathbb{N}$ then $\displaystyle T$ are all the elements of $\displaystyle S$ negated...

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

    then

    $\displaystyle 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 $\displaystyle S \subset \mathbb{N}$ then it can't be a subset of the negative integers.
    Quote Originally Posted by novice View Post
    Given a set of negative integers, $\displaystyle S$, where $\displaystyle S \subset \mathbb{Z}^-$, we can build a set $\displaystyle T$ of positive numbers by a set builder notation such as
    $\displaystyle T=\{x \in Z: -x \in S\}$
    Did you bother to read the OP?
    It clearly states that $\displaystyle S$ is a set of negative integers.
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    Quote Originally Posted by gmatt View Post
    I'm not exactly sure what you mean, if $\displaystyle S \subset \mathbb{N}$ 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 $\displaystyle S$, not adding another condition onto it ( since trivially if $\displaystyle S \subset \mathbb{N}$ and $\displaystyle S \subset \mathbb{Z}^-$ then $\displaystyle S = \emptyset$)
    Sorry, I didn't mean to confuse anyone. I had one thing and I asked myself a question by switching $\displaystyle S\subset \mathbb{Z}^-$ to $\displaystyle S\subset \mathbb{Z}^+$. 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 $\displaystyle S$ 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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