**determine whether {2} and 2 is an element of {{2}, {2,{2}}**

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$Solution: ~Given~\{\{2\},\{2,\{ 2\}\}\} \\ For~ element ~\{2\}:The ~set ~has ~two ~elements. One ~of ~them ~is ~patently~\{2\}. \\\{2\} ~is ~ a ~ subset. \\\therefore \{2\}~ is ~ an ~element~ of ~ the ~set. \\ For~ element ~2:The ~ set~ contains ~ the ~subset~\{\{ 2\}\}~(subset~ containing~ a~ subset). \\The ~set ~contains~ only~ subsets ,while~2 ~is ~not~ a ~ subset~ and~ thus~ 2~ is ~not~ an~ element~ \\of ~the~ set. \\\therefore 2~ is~ not ~ an ~element~ of ~ the ~set.$