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Math Help - Set Equality and Cardinality

  1. #1
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    Set Equality and Cardinality

    Let me define two sets:
    A={1}
    B={1,1}

    Few questions:
    1. Is A=B?
    2. How many elements (or cardinality) does A have? What about B?
    3. Is there any time in mathematics that we need to make a distinction between A and B?

    (I guess I am just confused, so any guidance would be welcome)
    Thanks,
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  2. #2
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    Quote Originally Posted by aman_cc View Post
    Let me define two sets:
    A={1}
    B={1,1}

    Few questions:
    1. Is A=B?

    \color{red}\mbox{Yes. In sets we have no multiplicity, so you can put 1 million 1's, it's the same as A}


    2. How many elements (or cardinality) does A have? What about B?

    \color{red}\mbox{as } A=B \,\,\,\,\mbox{then...}


    3. Is there any time in mathematics that we need to make a distinction between A and B?

    \color{red}\mbox{As sets and in the usual frame of set theory axioms the answer is no}

    \color{red} Tonio

    (I guess I am just confused, so any guidance would be welcome)
    Thanks,
    .
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  3. #3
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    A set is complelety defined by its elements, and since same name means same object (I guess it is quite reasonnable in mathematics): in \{1,1\}, there is only one thing, which is 1, thus \{1,1\}=\{1\}.

    When you want to consider a collection of objects and be able to see them appear multiple times, you can use families or sequences or tuples.

    EDIT: Ah Tonio was faster
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  4. #4
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    Quote Originally Posted by clic-clac View Post
    A set is complelety defined by its elements, and since same name means same object (I guess it is quite reasonnable in mathematics): in \{1,1\}, there is only one thing, which is 1, thus \{1,1\}=\{1\}.

    When you want to consider a collection of objects and be able to see them appear multiple times, you can use families or sequences or tuples.

    EDIT: Ah Tonio was faster
    Thanks clic-clac and Tonio. Really appreciate your help.
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  5. #5
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    3. Is there any time in mathematics that we need to make a distinction between A and B?
    Yes, in multisets: Multiset - Wikipedia, the free encyclopedia, but I have never really worked explicitly with them.
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  6. #6
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    Quote Originally Posted by aman_cc View Post
    3. Is there any time in mathematics that we need to make a distinction between A and B?
    There is also something related called the disjoint union. It indexes elements based on which sets they came from. For instance, if A_1=A_2=\{1\},

    A_1\cup A_2=\{1\} (regular union)

    A_1\sqcup A_2=\{(1,1),(1,2)\} (disjoint union)

    (Note that those should be different union symbols, though they look sort of similar.)
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