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Math Help - Transitive set

  1. #1
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    Transitive set

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

    For the other direction. I need to show that A \subseteq PA. I have a hard time of effectively using the hypothesis that 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 A is a transitive set if and only if PA is a transitive set.
    I think I was able to prove one direction.
    Suppose A is transitive. We need to show that PA \subseteq PPA. Let x \in PA. Then x \subseteq A. Since A is transitive, we know A\subseteq PA. Hence, x \subseteq PA \Rightarrow x \in PPA.

    For the other direction. I need to show that A \subseteq PA. I have a hard time of effectively using the hypothesis that 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. x \in X \rightarrow x \subseteq X.
    Lemma 2. X \subseteq PX.

    Suppose PA is a transitive set.

    If A \in PA, then A \subseteq PA by Lemma 1.
    If A \subseteq PA, then A is a transitive set by Lemma 2.
    Last edited by aliceinwonderland; February 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. x \in X \rightarrow x \subseteq X.
    Lemma 2. X \subseteq PX.

    Suppose PA is a transitive set.

    If A \in PA, then A \subseteq PA by Lemma 1.
    If 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 A is transitive then \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 A is also an element in PA by the usual way, but I couldn't get anywhere.
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