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Thread: quick question about set

  1. #1
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    quick question about set

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

    Thank you
    Last edited by zpwnchen; Sep 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. $\displaystyle \emptyset \subset ${0} <-- True True, the empty set is a subset of every set
    b. $\displaystyle \emptyset \subset ${$\displaystyle \emptyset $,{0}} <-- True True
    c. {$\displaystyle \emptyset $}$\displaystyle \in ${$\displaystyle \emptyset$} <-- True No, it's false. $\displaystyle \color{red}\{\emptyset\}=\{\emptyset\}$
    d. { $\displaystyle \emptyset$ } $\displaystyle \in$ {{ $\displaystyle \emptyset$ }} <-- False No, it's true. It is always true that $\displaystyle \color{red}\{x\}\in\{\{x\}\}$ whatever $\displaystyle \color{red}x$ may stand for.
    e. {$\displaystyle \emptyset $} $\displaystyle \subset $ {$\displaystyle \emptyset $ , {$\displaystyle \emptyset $}} <-- True True
    f. {{$\displaystyle \emptyset $}} $\displaystyle \subset $ {$\displaystyle \emptyset $,{$\displaystyle \emptyset $}} <--False No, it's true: $\displaystyle \color{red}\{\{x\}\}\subset\{\emptyset,\{x\}\}$
    g. {{$\displaystyle \emptyset $}} $\displaystyle \subset $ {{$\displaystyle \emptyset $},{$\displaystyle \emptyset $}} <--False Correct, it is false, it should be $\displaystyle \color{red}\{\{\emptyset\}\}\subseteq\{\{\emptyset \},\{\emptyset\}\}$ or $\displaystyle \color{red}\{\{\emptyset\}\}=\{\{\emptyset\},\{\em ptyset\}\}$

    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. {{$\displaystyle \emptyset $}} $\displaystyle \subset $ {{$\displaystyle \emptyset $},{$\displaystyle \emptyset $}} <--False

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

    The answer depends on your interpretation of the symbol $\displaystyle \subset$. Some authors define this symbol to mean either proper inclusion or equality of sets. I.e., $\displaystyle A \subset B$ 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 $\displaystyle \subset$ to mean proper inclusion, i.e. $\displaystyle A \subset B$ means every element of A is also an element of B but $\displaystyle A \neq B$, and write $\displaystyle A \subseteq B$ or $\displaystyle A \subseteqq B$ 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
    $\displaystyle
    \color{red}\{\{\emptyset\}\}=\{\{\emptyset\},\{\em ptyset\}\}
    $
    $\displaystyle
    \color{red}\{\{\emptyset\}\}\subseteq\{\{\emptyset \},\{\emptyset\}\}
    $

    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 $\displaystyle
    A \neq B
    $ right... ?sorry a little bit confused
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  5. #5
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    Quote Originally Posted by zpwnchen View Post
    $\displaystyle
    \color{red}\{\{\emptyset\}\}=\{\{\emptyset\},\{\em ptyset\}\}
    $
    $\displaystyle
    \color{red}\{\{\emptyset\}\}\subseteq\{\{\emptyset \},\{\emptyset\}\}
    $

    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 $\displaystyle
    A \neq B$
    By definition for sets $\displaystyle P=Q$ if and only if $\displaystyle P \subseteq Q\;\& \;Q \subseteq P$.

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

    So $\displaystyle \{\{\emptyset\}\}=\{\{\emptyset\},\{\emptyset\}\}
    $



    BTW: The set $\displaystyle \{a,a\}$ 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 $\displaystyle P=Q$ if and only if $\displaystyle P \subseteq Q\;\& \;Q \subseteq P$.

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

    So $\displaystyle \{\{\emptyset\}\}=\{\{\emptyset\},\{\emptyset\}\}
    $


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

    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
    $\displaystyle \emptyset \in \{x\} $ False
    $\displaystyle \emptyset \subset \{x\}$ True
    $\displaystyle \emptyset \subseteq \{x\}$ 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 $\displaystyle x \ne \emptyset$
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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 $\displaystyle \subset$ B and A $\displaystyle \subseteq$ B? But A $\displaystyle \ne$ 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 $\displaystyle \subset$ B and A $\displaystyle \subseteq$ B? But A $\displaystyle \ne$ B, right?
    Well $\displaystyle A \not\subset B$. A contains elements that are not in B.
    But $\displaystyle B \subset A$, 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 $\displaystyle \subset$ A and B $\displaystyle \subseteq$ A? But A $\displaystyle \ne$ B, right?
    YES
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