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Math Help - Equality Question

  1. #1
    No one in Particular VonNemo19's Avatar
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    Equality Question

    Why is it the case that \emptyset=\{\emptyset\},

    and also that

    \{1,2,3\}=\{\{1\},\{2\},\{3\}\} ?

    I'm having trouble proving these statements from the definition.

    Let x\in\emptyset... OK, now I'm lost.
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  2. #2
    MHF Contributor
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    Both of these statements are false. Sets are equal iff they have the same elements. The empty set \emptyset does not have any elements whereas \{\emptyset\} has one element \emptyset. Similarly, 1\ne\{1\}, 1\ne\{2\}, 1\ne\{3\}, so 1\in\{1,2,3\} but 1\notin\{\{1\},\{2\},\{3\}\}.

    Maybe there is something in the context that would make sense of these equalities?
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  3. #3
    No one in Particular VonNemo19's Avatar
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    Thanks for the reply. I should have specified that the problems were to determine whether or not the sets were equal. Sorry for the confusion.
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