criterion for a set to be transitive

Theorem.

A set $X$ is transitive if and only if its power set $\\mathcal{P}(X)$ is transitive.

Proof.

First assume $X$ is transitive. Let $A\\in B\\in\\mathcal{P}(X)$ . Since $B\\in\\mathcal{P}(X)$ , $B\\subseteq X$ . Thus, $A\\in X$ . Since $X$ is transitive, $A\\subseteq X$ . Hence, $A\\in\\mathcal{P}(X)$ . It follows that $\\mathcal{P}(X)$ is transitive.

Conversely, assume $\\mathcal{P}(X)$ is transitive. Let $a\\in X$ . Then $\\{a\\}\\in\\mathcal{P}(X)$ . Since $\\mathcal{P}(X)$ is transitive, $\\{a\\}\\subseteq\\mathcal{P}(X)$ . Thus, $a\\in\\mathcal{P}(X)$ . Hence, $a\\subseteq X$ . It follows that $X$ is transitive.∎

单词	CriterionForASetToBeTransitive
释义	criterion for a set to be transitive Theorem. A set $X$ is transitive if and only if its power set $\\mathcal{P}(X)$ is transitive. Proof. First assume $X$ is transitive. Let $A\\in B\\in\\mathcal{P}(X)$ . Since $B\\in\\mathcal{P}(X)$ , $B\\subseteq X$ . Thus, $A\\in X$ . Since $X$ is transitive, $A\\subseteq X$ . Hence, $A\\in\\mathcal{P}(X)$ . It follows that $\\mathcal{P}(X)$ is transitive. Conversely, assume $\\mathcal{P}(X)$ is transitive. Let $a\\in X$ . Then $\\{a\\}\\in\\mathcal{P}(X)$ . Since $\\mathcal{P}(X)$ is transitive, $\\{a\\}\\subseteq\\mathcal{P}(X)$ . Thus, $a\\in\\mathcal{P}(X)$ . Hence, $a\\subseteq X$ . It follows that $X$ is transitive.∎
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