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Math Help - [SOLVED] How Much of a Proof is this

  1. #1
    Member billa's Avatar
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    [SOLVED] How Much of a Proof is this

    <br />
A\subseteq B : given

    <br />
R[A]=\{y\mid(\exists x\in A)(x,y)\in R\}<br />
    <br />
R[ B ]=\{y\mid(\exists x\in B)(x,y)\in R\}<br />
    (\forall x\in A)\,\, x\in B
    <br />
(\forall y\in R[A])\,\, y\in\{y\mid(\exists x\in B)\,(x,y)\in R\}<br />

    <br />
R[A]\subseteq R[ B ]<br />

    I have to prove the last line in that list; does this constitute a proof?

    Thanks, I am unused to both set theory and proofs, which makes this especially hard.
    Last edited by billa; July 11th 2009 at 01:53 PM. Reason: missspellingg
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  2. #2
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    Quote Originally Posted by billa View Post
    <br />
A\subseteq B : given

    <br />
R[A]=\{y\mid(\exists x\in A)(x,y)\in R\}<br />
    <br />
R[ B ]=\{y\mid(\exists x\in B)(x,y)\in R\}<br />
    (\forall x\in A)\,\, x\in B
    <br />
(\forall y\in R[A])\,\, y\in\{y\mid(\exists x\in B)\,(x,y)\in R\}<br />

    <br />
R[A]\subseteq R[ B ]<br />

    I have to proof the last line in that list; does this constitute a proof?

    Thanks, I am unused to both set theory and proofs, which makes this especially hard.
    \begin{gathered}<br />
  q \in R\left[ A \right]\; \Rightarrow \;\left( {\exists p \in A} \right)\left[ {\left( {p,q} \right) \in R} \right] \hfill \\<br />
  A \subseteq B\; \Rightarrow \;p \in B\; \Rightarrow \;\left( {q \in R\left[ B \right]} \right) \hfill \\<br />
  R\left[ A \right] \subseteq R\left[ B \right] \hfill \\ <br />
\end{gathered}
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  3. #3
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    Quote Originally Posted by Plato View Post
    <br /> <br />
  A \subseteq B\; \Rightarrow \;p \in B\; \Rightarrow \;\left( {q \in R\left[ B \right]} \right) \hfill \\<br /> <br />
    excuse me, but how did you get that?
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  4. #4
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    Quote Originally Posted by doresa View Post
    excuse me, but how did you get that?
    Well it is very clear.
    We know that (p,q)\in R \text{ and } p\in B so by definition q\in R[ B ].
    Last edited by Plato; July 12th 2009 at 11:35 AM.
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