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Thread: Transitive set

  1. #1
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    Transitive set

    Prove that $\displaystyle A$ is a transitive set if and only if $\displaystyle PA$ is a transitive set.
    I think I was able to prove one direction.
    Suppose $\displaystyle A$ is transitive. We need to show that $\displaystyle PA \subseteq PPA$. Let $\displaystyle x \in PA$. Then $\displaystyle x \subseteq A$. Since $\displaystyle A$ is transitive, we know $\displaystyle A\subseteq PA$. Hence, $\displaystyle x \subseteq PA \Rightarrow x \in PPA$.

    For the other direction. I need to show that $\displaystyle A \subseteq PA$. I have a hard time of effectively using the hypothesis that $\displaystyle PA$ is a transitive set. Can someone give me a hand here? Thanks.
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  2. #2
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    Quote Originally Posted by namelessguy View Post
    Prove that $\displaystyle A$ is a transitive set if and only if $\displaystyle PA$ is a transitive set.
    I think I was able to prove one direction.
    Suppose $\displaystyle A$ is transitive. We need to show that $\displaystyle PA \subseteq PPA$. Let $\displaystyle x \in PA$. Then $\displaystyle x \subseteq A$. Since $\displaystyle A$ is transitive, we know $\displaystyle A\subseteq PA$. Hence, $\displaystyle x \subseteq PA \Rightarrow x \in PPA$.

    For the other direction. I need to show that $\displaystyle A \subseteq PA$. I have a hard time of effectively using the hypothesis that $\displaystyle PA$ is a transitive set. Can someone give me a hand here? Thanks.
    If X is a transitive set, then the followings hold (You need to verify the below lemmas).

    Lemma 1. $\displaystyle x \in X \rightarrow x \subseteq X$.
    Lemma 2. $\displaystyle X \subseteq PX$.

    Suppose PA is a transitive set.

    If $\displaystyle A \in PA$, then $\displaystyle A \subseteq PA$ by Lemma 1.
    If $\displaystyle A \subseteq PA$, then A is a transitive set by Lemma 2.
    Last edited by aliceinwonderland; Feb 24th 2009 at 02:06 AM.
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  3. #3
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    Quote Originally Posted by aliceinwonderland View Post
    If X is a transitive set, then the followings hold (You need to verify the below lemmas).

    Lemma 1. $\displaystyle x \in X \rightarrow x \subseteq X$.
    Lemma 2. $\displaystyle X \subseteq PX$.

    Suppose PA is a transitive set.

    If $\displaystyle A \in PA$, then $\displaystyle A \subseteq PA$ by Lemma 1.
    If $\displaystyle A \subseteq PA$, then A is a transitive set by Lemma 2.
    Thank you very much for your help Alice. We did prove the lemmas in class, one more lemma we proved is that if $\displaystyle A$ is transitive then $\displaystyle \bigcup A \subseteq A$. I didn't approach this problem as you did, which is why I was stuck I think. I was trying to show that any element in $\displaystyle A$ is also an element in $\displaystyle PA$ by the usual way, but I couldn't get anywhere.
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