Assume that P(A) = P(B). Show that A = B. | vedclass

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Given that $P(A) = P(B)$.To show: $A = B$.Let $x \in A$.Since $A \in P(A)$ and $P(A) = P(B)$,it follows that $A \in P(B)$.By the definition of the pow

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Given that $P(A) = P(B)$.To show: $A = B$.Let $x \in A$.Since $A \in P(A)$ and $P(A) = P(B)$,it follows that $A \in P(B)$.By the definition of the power set,if $A \in P(B)$,then $A \subseteq B$.Therefore,$x \in A \implies x \in B$,which means $A \subseteq B$.Similarly,let $y \in B$. Since $B \in P(B)$ and $P(B) = P(A)$,it follows that $B \in P(A)$.By the definition of the power set,if $B \in P(A)$,then $B \subseteq A$.Therefore,$y \in B \implies y \in A$,which means $B \subseteq A$.Since $A \subseteq B$ and $B \subseteq A$,we conclude that $A = B$.

Assume that $P(A) = P(B)$. Show that $A = B$.

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