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Thread: Equality Question

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

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

    and also that

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

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

    Let $\displaystyle x\in\emptyset$... 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 $\displaystyle \emptyset$ does not have any elements whereas $\displaystyle \{\emptyset\}$ has one element $\displaystyle \emptyset$. Similarly, $\displaystyle 1\ne\{1\}$, $\displaystyle 1\ne\{2\}$, $\displaystyle 1\ne\{3\}$, so $\displaystyle 1\in\{1,2,3\}$ but $\displaystyle 1\notin\{\{1\},\{2\},\{3\}\}$.

    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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