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$\langle x,y \rangle$ is a Kuratowski pair. Prove that $\Bigl(\cap\cup\langle x,y \rangle\Bigr) \bigcup \Bigl((\cup\cup\langle x,y \rangle)\setminus(\cup\cap\langle x,y \rangle)\Bigr)=y$

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    You should, if so, write an answer to your own question and if it is indeed correct you can accept it.2012-01-25

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Note that $\begin{align*} \cup\langle x,y\rangle &= \cup\{ \{x\},\{x,y\}\}= \{x\}\bigcup\{x,y\} = \{x,y\}\\ \cap\langle x,y\rangle &= \cap\{ \{x\},\{x,y\}\} = \{x\}\bigcap\{x,y\}=\{x\}. \end{align*}$

So $\begin{align*} \cap\cup\langle x,y\rangle &= \cap\{x,y\} = x\cap y,\\ \cup\cap\langle x,y\rangle &= \cup\{x\} = x,\\ \cup\cup\langle x,y\rangle &= \cup\{x,y\} = x\cup y. \end{align*}$

Things should be rather easy now.