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.∎
随便看	dual group of G is homeomorphic to the character space of L^1⁢(G) DualHomomorphism dual homomorphism DualHomomorphismOfTheDerivative dual homomorphism of the derivative DualIsogeny dual isogeny DualityInMathematics duality in mathematics DualityOfGudermannianAndItsInverseFunction duality of Gudermannian and its inverse function DualityWithRespectToANondegenerateBilinearForm duality with respect to a non-degenerate bilinear form DualModule dual module DualOfACoalgebraIsAnAlgebraThe dual of a coalgebra is an algebra, the DualOfDilworthsTheorem dual of Dilworth’s theorem DualOfStoneRepresentationTheorem dual of Stone representation theorem DualSpace dual space DualSpaceOfABooleanAlgebra dual space of a Boolean algebra