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Thread: A contradiction about sets!

  1. #1
    Junior Member Researcher's Avatar
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    Exclamation A contradiction about sets!

    Hello all.
    I know the last expresion is not allways true(by making examples), but it can be proved that is true!

    $\displaystyle x \in ( A \cap B) \equiv [( x \in A ) \wedge ( x \in B )]$

    We know that to being the $\displaystyle x \in ( A \cap B)$ a true expression, both the expressions right to the (first) equalency sign must be true. So we have:

    $\displaystyle x \in ( A \cap B) \equiv [ ( x \in A ) \wedge ( x \in B) ] \equiv [ [ ( x \in A ) \wedge T ] \equiv ( x \in A ) ] \Rightarrow ( A \cap B ) = A$

    It can be proved by a similar way that: $\displaystyle ( A \cap B ) = B$

    -------------------------
    What do examples contradict to this theorem?
    Does the proof have a problem?

    Thanks.
    Last edited by Researcher; Sep 8th 2009 at 02:09 PM.
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  2. #2
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    Quote Originally Posted by Researcher View Post
    Hello all.
    I know the last expresion is not allways true(by making examples), but it can be proved that is true!

    $\displaystyle x \in a ( A \cap B) \equiv [( x \in A ) \wedge ( x \in B )]$

    We know that to being the $\displaystyle x \in ( A \cap B)$ a true expression, both the expressions right to the (first) equalency sign must be true. So we have:

    $\displaystyle x \in ( A \cap B) \equiv [ ( x \in A ) \wedge ( x \in B) ] \equiv [ [ ( x \in A ) \wedge T ] \equiv ( x \in A ) ] \Rightarrow ( A \cap B ) = A$

    It can be proved by a similar way that: $\displaystyle ( A \cap B ) = B$

    -------------------------
    What do examples contradict to this theorem?
    Does the proof have a problem?

    Thanks.
    Yes, there is a problem with the "theorem"! You have proved that "if $\displaystyle x\in A\cap B$ then $\displaystyle x\in A$". That only proves $\displaystyle A\cap B\subset A$. In order to prove $\displaystyle A\cap B= A$, you would have to prove "if $\displaystyle x\in A$ then $\displaystyle x\in A\cap B$" which you can't do.
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    Yes, there is a problem with the "theorem"! You have proved that "if $\displaystyle x\in A\cap B$ then $\displaystyle x\in A$". That only proves $\displaystyle A\cap B\subset A$. In order to prove $\displaystyle A\cap B= A$, you would have to prove "if $\displaystyle x\in A$ then $\displaystyle x\in A\cap B$" which you can't do.
    No you are wrong!
    Because by propositional calculus we know that $\displaystyle ( P \wedge T ) \equiv P$. And We know that $\displaystyle P \equiv Q$ is the same (P => Q and Q => P).
    (So i have proved that [($\displaystyle x\in A\cap B$ then $\displaystyle x\in A$) And ($\displaystyle x\in A$ then $\displaystyle x\in A\cap B$)]
    And we know from The Axiom of Extensoinality:
    If for every x, $\displaystyle [( x\in A ) \equiv ( x\in B )] \Rightarrow A = B$

    You can see that i have proved that $\displaystyle [x \in ( A \cap B) \equiv ( x \in A )] \Rightarrow ( A \cap B ) = A$
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    Quote Originally Posted by Researcher View Post
    No you are wrong!
    Because by propositional calculus we know that $\displaystyle ( P \wedge T ) \equiv P$. And We know that $\displaystyle P \equiv Q$ is the same (P => Q and Q => P).
    (So i have proved that [($\displaystyle x\in A\cap B$ then $\displaystyle x\in A$) And ($\displaystyle x\in A$ then $\displaystyle x\in A\cap B$)]
    And we know from The Axiom of Extensoinality:
    If for every x, $\displaystyle [( x\in A ) \equiv ( x\in B )] \Rightarrow A = B$

    You can see that i have proved that $\displaystyle [x \in ( A \cap B) \equiv ( x \in A )] \Rightarrow ( A \cap B ) = A$

    Take A = { 1 } ,B = { 2} ,THEN $\displaystyle A\cap B\neq A$

    So your theorem is not correct
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  5. #5
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    I said in my first post that there are examples wich contradict with this theorem.
    But my question is: Why by using the same valid logical expresion(wich are also used to buld all the bases of set theory)does this theorem appear?
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  6. #6
    MHF Contributor Matt Westwood's Avatar
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    By propositional calculus you indeed have that $\displaystyle P \land Q \implies P$ but it is NOT true that $\displaystyle P \implies P \land Q$.

    So where you have (in your original expression): $\displaystyle [[x \in A \land T] \equiv x \in A]$ the equivalence does not hold because you do not have $\displaystyle [x \in A \implies [x \in A \land T]]$.


    That's where you have gone wrong.
    Last edited by Matt Westwood; Sep 8th 2009 at 10:42 PM. Reason: amplifying
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  7. #7
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    Quote Originally Posted by Matt Westwood View Post
    By propositional calculus you indeed have that $\displaystyle P \land Q \implies P$ but it is NOT true that $\displaystyle P \implies P \land Q$.

    So where you have (in your original expression): $\displaystyle [[x \in A \land T] \equiv x \in A]$ the equivalence does not hold because you do not have $\displaystyle [x \in A \implies [x \in A \land T]]$.


    That's where you have gone wrong.
    No, you are wrong!!
    Your statement ( $\displaystyle P \implies P \land Q$ is not true) holds whenever we don't know about validity of Q.(i mean we don't know it is true or false).
    But i've said clearly that $\displaystyle Q = T$ here (wich $\displaystyle T$ is a true statement, and here it is $\displaystyle x \in B$ ).
    So the equvalency sign is i have chosen is TRUE.
    Because $\displaystyle P \land T \implies P$ and $\displaystyle P \implies P \land T$.
    Last edited by Researcher; Sep 8th 2009 at 11:18 PM.
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    Quote Originally Posted by Researcher View Post
    Hello all.
    I know the last expresion is not allways true(by making examples), but it can be proved that is true!

    $\displaystyle x \in ( A \cap B) \equiv [( x \in A ) \wedge ( x \in B )]$

    We know that to being the $\displaystyle x \in ( A \cap B)$ a true expression, both the expressions right to the (first) equalency sign must be true. So we have:

    $\displaystyle x \in ( A \cap B) \equiv [ ( x \in A ) \wedge ( x \in B) ] \equiv [ [ ( x \in A ) \wedge T ] \equiv ( x \in A ) ] \Rightarrow ( A \cap B ) = A$

    It can be proved by a similar way that: $\displaystyle ( A \cap B ) = B$

    -------------------------
    What do examples contradict to this theorem?
    Does the proof have a problem?

    Thanks.
    First of all we cannot mix a syntactical with semantical proof ,a thing that you have done in your proof.

    In a syntactical proof one uses the rules of logic without using true or false values for parts of formulas.

    In a semantical proof one uses the true tables .

    Mixing the two the results are wrong conclusions.

    Besides that, to have: $\displaystyle [ ( x \in A ) \wedge ( x \in B) ] \equiv [ [ ( x \in A ) \wedge T ]$ we must have that :

    $\displaystyle (x\in A)\wedge T) $ logicaly implies $\displaystyle (x\in A)\wedge (x\in B)$

    ................................................OR.......................

    The conditional : $\displaystyle [(x\in A)\wedge T]\Longrightarrow [(x\in A)\wedge (x\in B)]$ is a tautology.


    But it is not
    .

    Hence : $\displaystyle [ ( x \in A ) \wedge ( x \in B) ] \equiv [ [ ( x \in A ) \wedge T ]$ does not hold .

    Thus your proof is wrong

    AND another more obvious thing: In the backwards proof ( the one starting with xεΑ ) When you substitute T = xεΒ ,this is wrong because you do not know now the value of xεΒ
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  9. #9
    MHF Contributor Matt Westwood's Avatar
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    Quote Originally Posted by Matt Westwood View Post
    By propositional calculus you indeed have that $\displaystyle P \land Q \implies P$ but it is NOT true that $\displaystyle P \implies P \land Q$.

    So where you have (in your original expression): $\displaystyle [[x \in A \land T] \equiv x \in A]$ the equivalence does not hold because you do not have $\displaystyle [x \in A \implies [x \in A \land T]]$.


    That's where you have gone wrong.
    I think I understand now (of course, "T" is "True" = "tautology")

    IF it is the case that $\displaystyle x \in B \equiv T$, which it only is if $\displaystyle x$ is ALWAYS an element of $\displaystyle B$, then the equivalence indeed holds. But then that means $\displaystyle A \subseteq B$ which is indeed equivalent to $\displaystyle A \cup B = A$.

    But first you have to take as a truth that $\displaystyle x \in B$ ALL THE TIME, WHATEVER $\displaystyle x$ and $\displaystyle B$ may be.

    WHICH IS RUBBISH!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
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