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Math Help - set of odd integers proof

  1. #1
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    set of odd integers proof

    I am working on a simple set theory proof involving the definition of odd numbers, and so far I've done one containment. I would guess that if thiss is correct, then the other containment would be equally simple. Does this look alright so far?

    \mbox{If }A=\{x \in \mathbb{Z}~|~x = 2k+1\mbox{ for some }k \in \mathbb{Z}\} and  B=\{y \in \mathbb{Z}~|~y=2s-1\mbox{ for some }s \in \mathbb{Z}\}, prove that A=B

    \mbox{\textbf{Proof.}} Let x\in A. then \exists~k \in \mathbb{Z}\mbox{ such that }x=2k+1. Equivalently,
    \Longrightarrow x=2k+1+1-1
    \Longrightarrow x=2k+2-1
    \Longrightarrow x=2(k+1)-1

    Since k \in\mathbb{Z} \Longrightarrow k+1 \in \mathbb{Z}  x = 2(k+1)-1 \Longrightarrow x \in B. Therefore, A\subseteq B
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  2. #2
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    Quote Originally Posted by james121515 View Post
    I am working on a simple set theory proof involving the definition of odd numbers, and so far I've done one containment. I would guess that if thiss is correct, then the other containment would be equally simple. Does this look alright so far?

    \mbox{If }A=\{x \in \mathbb{Z}~|~x = 2k+1\mbox{ for some }k \in \mathbb{Z}\} and  B=\{y \in \mathbb{Z}~|~y=2s-1\mbox{ for some }s \in \mathbb{Z}\}, prove that A=B

    \mbox{\textbf{Proof.}} Let x\in A. then \exists~k \in \mathbb{Z}\mbox{ such that }x=2k+1. Equivalently,
    \Longrightarrow x=2k+1+1-1
    \Longrightarrow x=2k+2-1
    \Longrightarrow x=2(k+1)-1

    Since k \in\mathbb{Z} \Longrightarrow k+1 \in \mathbb{Z}  x = 2(k+1)-1 \Longrightarrow x \in B. Therefore, A\subseteq B
    That 'way' is correct.
    By symmetry you are done.

    x=2k+1=2(k+1)-1
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  3. #3
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    Thanks for your response.

    So you are saying that due to symmetry, there is no need to show the other "right to left" containment due to symmetry?

    -James
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    Quote Originally Posted by james121515 View Post
    Thanks for your response.

    So you are saying that due to symmetry, there is no need to show the other "right to left" containment due to symmetry?
    x=2k+1=2(k+1)-1
    There is the symmetry.
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