cindylee2016
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- Sep 7, 2016
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\(\displaystyle \mbox{Let }\, A\, \mbox{ be a nonempty set and }\, 2^A\, \mbox{ is its power set. Fix }\, S\, \subset\, A\, \mbox{ and}\)
\(\displaystyle \mbox{define }\, F\, :\, 2^A\, \rightarrow\, 2^A\, \mbox{ by}\)
. . . . .\(\displaystyle F(X)\, =\, X\, \bigcap\, A\, \mbox{ for all }\, X\, \in\, 2^A.\)
\(\displaystyle \mbox{Prove that }\, F\, \mbox{ is injective if and only if it is surjective. Find }\, S\, \mbox{ if }\, F\)
\(\displaystyle \mbox{is indeed injective.}\)
\(\displaystyle \mbox{define }\, F\, :\, 2^A\, \rightarrow\, 2^A\, \mbox{ by}\)
. . . . .\(\displaystyle F(X)\, =\, X\, \bigcap\, A\, \mbox{ for all }\, X\, \in\, 2^A.\)
\(\displaystyle \mbox{Prove that }\, F\, \mbox{ is injective if and only if it is surjective. Find }\, S\, \mbox{ if }\, F\)
\(\displaystyle \mbox{is indeed injective.}\)
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