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Math Help - Set Theory Proof

  1. #1
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    Set Theory Proof

    Prove { {a}, {a,b} } = { {c}, {c,d} } if and only if a=b and c=d.

    I'm wondering how to proceed. I'm not even sure what exactly am I trying to prove
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  2. #2
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    Quote Originally Posted by TitaniumX View Post
    Prove { {a}, {a,b} } = { {c}, {c,d} } if and only if a=b and c=d.

    I'm wondering how to proceed. I'm not even sure what exactly am I trying to prove

    For sets A,B the definition tells us that A=B\Longleftrightarrow A\subset B\,\,\,and\,\,\,B\subset A , so:

    \{ \{a\}, \{a,b\} \} = \{ \{c\}, \{c,d\} \} \Longrightarrow \{ \{a\}, \{a,b\} \} \subset \{ \{c\}, \{c,d\} \} \Longrightarrow \{a\}\in \{ \{c\}, \{c,d\} \} , and so \{a\}=\{c\}\,\,\,or\,\,\,\{a\}=\{c,d\} .

    But \{a\} contains one single element so this set cannot be equal to a set containing two elements, and thus \{a\}=\{c\}\Longleftrightarrow a=c . Take it from here.

    Tonio
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