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Math Help - quick question about set

  1. #1
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    quick question about set

    Can you please check if they are correct?
    a. \emptyset \subset {0} <-- True
    b. \emptyset \subset {  \emptyset ,{0}} <-- True
    c. {  \emptyset }  \in { \emptyset} <-- True
    d. { \emptyset } \in {{ \emptyset }} <-- False
    e. {  \emptyset }  \subset {  \emptyset , {  \emptyset }} <-- True
    f. {{  \emptyset }}  \subset {  \emptyset ,{  \emptyset }} <--False
    g. {{  \emptyset }}  \subset {{  \emptyset },{  \emptyset }} <--False

    Thank you
    Last edited by zpwnchen; September 18th 2009 at 08:28 PM.
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  2. #2
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    Hello zpwnchen
    Quote Originally Posted by zpwnchen View Post
    Can you please check if they are correct?
    a. \emptyset \subset {0} <-- True True, the empty set is a subset of every set
    b. \emptyset \subset {  \emptyset ,{0}} <-- True True
    c. {  \emptyset }  \in { \emptyset} <-- True No, it's false. \color{red}\{\emptyset\}=\{\emptyset\}
    d. { \emptyset } \in {{ \emptyset }} <-- False No, it's true. It is always true that \color{red}\{x\}\in\{\{x\}\} whatever \color{red}x may stand for.
    e. {  \emptyset }  \subset {  \emptyset , {  \emptyset }} <-- True True
    f. {{  \emptyset }}  \subset {  \emptyset ,{  \emptyset }} <--False No, it's true: \color{red}\{\{x\}\}\subset\{\emptyset,\{x\}\}
    g. {{  \emptyset }}  \subset {{  \emptyset },{  \emptyset }} <--False Correct, it is false, it should be \color{red}\{\{\emptyset\}\}\subseteq\{\{\emptyset  \},\{\emptyset\}\} or \color{red}\{\{\emptyset\}\}=\{\{\emptyset\},\{\em  ptyset\}\}

    Thank you
    Grandad
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  3. #3
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    Quote Originally Posted by zpwnchen View Post
    Can you please check if they are correct?
    [snip]
    g. {{  \emptyset }}  \subset {{  \emptyset },{  \emptyset }} <--False

    Thank you
    Just a comment on g. and Grandad's answer--

    The answer depends on your interpretation of the symbol \subset. Some authors define this symbol to mean either proper inclusion or equality of sets. I.e., A \subset B means every element of A is also an element of B. If your book or teacher defines it this way then your answer is wrong.

    Other authors define \subset to mean proper inclusion, i.e. A \subset B means every element of A is also an element of B but A \neq B, and write A \subseteq B or A \subseteqq B if A may be a proper subset of B or equal to B. So it depends.
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  4. #4
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    Quote Originally Posted by Grandad View Post
    Hello zpwnchen

    Grandad
    <br />
\color{red}\{\{\emptyset\}\}=\{\{\emptyset\},\{\em  ptyset\}\}<br />
    <br />
\color{red}\{\{\emptyset\}\}\subseteq\{\{\emptyset  \},\{\emptyset\}\}<br />

    Why they are equal? there are two elements in B, one in A. so it means every elements of A is not every elements of B...so should be <br />
A \neq B<br />
right... ?sorry a little bit confused
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  5. #5
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    Quote Originally Posted by zpwnchen View Post
    <br />
\color{red}\{\{\emptyset\}\}=\{\{\emptyset\},\{\em  ptyset\}\}<br />
    <br />
\color{red}\{\{\emptyset\}\}\subseteq\{\{\emptyset  \},\{\emptyset\}\}<br />

    Why they are equal? there are two elements in B, one in A. so it means every elements of A is not every elements of B...so should be <br />
A \neq B
    By definition for sets P=Q if and only if P \subseteq Q\;\& \;Q \subseteq P.

    Therefore \{a\}=\{a,a\} because \{a\}\subseteq \{a,a\}\;\& \;\{a,a\} \subseteq \{a\}

    So \{\{\emptyset\}\}=\{\{\emptyset\},\{\emptyset\}\}<br />



    BTW: The set \{a,a\} has only one element.
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  6. #6
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    Quote Originally Posted by Grandad View Post
    Hello zpwnchen

    Grandad
    Quote Originally Posted by Plato View Post
    By definition for sets P=Q if and only if P \subseteq Q\;\& \;Q \subseteq P.

    Therefore \{a\}=\{a,a\} because \{a\}\subseteq \{a,a\}\;\& \;\{a,a\} \subseteq \{a\}

    So \{\{\emptyset\}\}=\{\{\emptyset\},\{\emptyset\}\}<br />


    BTW: The set \{a,a\} has only one element.
    Thank you
    \emptyset \in \{x\}
    \emptyset \subset \{x\}
    \emptyset \subseteq \{x\}

    can you please explain me why these are true or false?
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  7. #7
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    Quote Originally Posted by zpwnchen View Post
    Thank you
    \emptyset \in \{x\} False
    \emptyset \subset \{x\} True
    \emptyset \subseteq \{x\} True
    can you please explain me why these are true or false?
    You, yourself, have noted that the emptyset is a subset of every set.
    But in the first case you have the element symbol and usually we understand that x \ne \emptyset
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  8. #8
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    Thank you!
    how about in this case:
    A = {0,1,2,3,4,5,6,7,8,9}
    B = {0,2,4,6,8}
    can we say that A \subset B and A \subseteq B? But A \ne B, right?
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  9. #9
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    Quote Originally Posted by zpwnchen View Post
    Thank you!
    how about in this case:
    A = {0,1,2,3,4,5,6,7,8,9}
    B = {0,2,4,6,8}
    can we say that A \subset B and A \subseteq B? But A \ne B, right?
    Well A \not\subset B. A contains elements that are not in B.
    But B \subset A, every element in B is in A.
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  10. #10
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    ops! i'm so sorry that i wrote it wrong.
    should be
    can we say that B \subset A and B \subseteq A? But A \ne B, right?
    YES
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