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Math Help - Transitivity

  1. #1
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    Transitivity

    Prove that if every X \in S is transitive, then \bigcup S is transitive.
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  2. #2
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    Quote Originally Posted by qwe123 View Post
    Prove that if every X \in S is transitive, then \bigcup S is transitive.
    Definition. A set A is said to be transitive set iff every member of a member of A is itself a member of A:
    x \in a \in A \Rightarrow x \in A.

    This condition can be stated as:
    a \in A \Rightarrow a \subseteq A.

    To show \bigcup S is transitive,
    x \in \bigcup S \Rightarrow x \in X \in S \Rightarrow x \subseteq X \in S (X is transitive set by hypothesis) \Rightarrow x \subseteq \bigcup S.

    Thus \bigcup S is transitive.
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