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Thread: I would like to know for example how I prove or disprove this sentence ?

  1. #1
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    I would like to know for example how I prove or disprove this sentence ?

    Hi,
    I am new here ,
    I have stuck to Proved or disproved This sentence :
    If for any A,B,C groups exist A\C=B\C And A ∩ C =B ∩ C , then A⊆ B.

    Thanks very much for help.
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    Re: I would like to know for example how I prove or disprove this sentence ?

    Quote Originally Posted by MathPro17 View Post
    I have stuck to Proved or disproved This sentence :
    If for any A,B,C groups exist A\C=B\C And A ∩ C =B ∩ C , then A⊆ B.
    Please, please carefully reread the above. I think that is not what you really mean.
    Do you really mean groups? Or is it sets or collections?

    Are you aware that $A\setminus C=A\cap C^c$ where $C^c$ is the complement of C. That is not usual group theory.

    You write prove: $A\subseteq B$ really? $A$ is a subset of $B~?$
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    Re: I would like to know for example how I prove or disprove this sentence ?

    Quote Originally Posted by Plato View Post
    Please, please carefully reread the above. I think that is not what you really mean.
    Do you really mean groups? Or is it sets or collections?

    Are you aware that $A\setminus C=A\cap C^c$ where $C^c$ is the complement of C. That is not usual group theory.

    You write prove: $A\subseteq B$ really? $A$ is a subset of $B~?$
    Hi,
    I mean Sets , sorry for mistake.
    Yes they ask for : "AB" ,
    How could I prove that ?

    Thanks.
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  4. #4
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    Re: I would like to know for example how I prove or disprove this sentence ?

    Let $x \in A$. If $x \in C$, then $x \in A \cap C = B\cap C$, so $x \in B$.
    If $x \notin C$, then $x \in A \setminus C = B\setminus C$, so $x \in B$.

    Since either $x \in C$ or $x \notin C$, this covers all possibilities, and $x \in B$ for any $x \in A$. This is the definition of a subset.
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    Re: I would like to know for example how I prove or disprove this sentence ?

    Quote Originally Posted by MathPro17 View Post
    Hi,
    I am new here ,
    I have stuck to Proved or disproved This sentence :
    If for any A,B,C sets exist A\C=B\C And A ∩ C =B ∩ C , then A⊆ B.
    We know that $A\setminus C=B\setminus C$ or $A\cap C^c=B\cap C^c$

    $ \begin{align*}A\cap C^c&=B\cap C^c \\\text{and given }A\cap C&=B\cap C\\\text{we have }(A\cap C)\cup(A\cap C^c)&=(B\cap C)\cup(B\cap C^c)\\A\cap(C\cup C^c)&=B\cap(C\cup C^c)\\A\cap~ \mathcal{U})&=B\cap~ \mathcal{U}\\A&=B \end{align*}$
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    Re: I would like to know for example how I prove or disprove this sentence ?

    Thanks very much for both of you for help ,
    I would like to know if I can add to your solves that I take some "x" the belongs to one side and to prove to the next side ?
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    Re: I would like to know for example how I prove or disprove this sentence ?

    Quote Originally Posted by MathPro17 View Post
    Thanks very much for both of you for help ,
    I would like to know if I can add to your solves that I take some "x" the belongs to one side and to prove to the next side ?
    To prove that $A \subseteq B$, you must show that for all $x \in A$, it is also true that $x \in B$.
    You can also show that $A = B$ (as Plato did), since $A=B \Longrightarrow (A\subseteq B \wedge B\subseteq A)$
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    Re: I would like to know for example how I prove or disprove this sentence ?

    Quote Originally Posted by SlipEternal View Post
    Let $x \in A$. If $x \in C$, then $x \in A \cap C = B\cap C$, so $x \in B$.
    If $x \notin C$, then $x \in A \setminus C = B\setminus C$, so $x \in B$.

    Since either $x \in C$ or $x \notin C$, this covers all possibilities, and $x \in B$ for any $x \in A$. This is the definition of a subset.
    I have another questions of this : "If for any A,B,C sets exist C\A=C\B And A ∩ C =B ∩ C , then A⊆ B."
    I have tried to solve like that :
    Let x∈A. If x∈C, then x∈A∩C=B∩C, so x∈B.
    If x∉C, then x!∈C∖A=C∖B.

    Is that Okay ? ( At begging I want to take x∈C but I need element from set A to show that exist : A⊆ B )

    Thanks
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    Re: I would like to know for example how I prove or disprove this sentence ?

    Quote Originally Posted by MathPro17 View Post
    I have another questions of this : "If for any A,B,C sets exist C\A=C\B And A ∩ C =B ∩ C , then A⊆ B."
    I have tried to solve like that :
    Let x∈A. If x∈C, then x∈A∩C=B∩C, so x∈B.
    If x∉C, then x!∈C∖A=C∖B.

    Is that Okay ? ( At begging I want to take x∈C but I need element from set A to show that exist : A⊆ B )

    Thanks
    No, that does not follow.

    Let $x \in A$ is the first step. Now, you have an element of $A$.

    Now, you have two cases:
    Case $x \in C$: Then $x \in A\cap C = B\cap C \subseteq B$ shows $x \in B$.
    Case $x \notin C$: Then $x \in A \setminus C = B\setminus C \subseteq B$ shows that $x \in B$.
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    Re: I would like to know for example how I prove or disprove this sentence ?

    Quote Originally Posted by MathPro17 View Post
    I have another questions of this : "If for any A,B,C sets exist C\A=C\B And A ∩ C =B ∩ C , then A⊆ B."
    I have tried to solve like that :
    In other words, you have now changed to $\large\color{red}{C\setminus A=C\setminus B}$ from $\large\color{blue}{A\setminus C=B\setminus C}$ with the same conclusion?

    If so consider: $A=\{p,q,x,w,y\}~,~B=\{r,s,x,w,y\},~\&~C=\{u,v,x,w ,y\}$ Can $A\subseteq B~?$
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    Re: I would like to know for example how I prove or disprove this sentence ?

    Quote Originally Posted by SlipEternal View Post
    No, that does not follow.

    Let $x \in A$ is the first step. Now, you have an element of $A$.

    Now, you have two cases:
    Case $x \in C$: Then $x \in A\cap C = B\cap C \subseteq B$ shows $x \in B$.
    Case $x \notin C$: Then $x \in A \setminus C = B\setminus C \subseteq B$ shows that $x \in B$.
    Thanks very much SlipEternal and Plato ,
    I have another difficult question that I don't know how to solve :
    "For any A,B,C Sets if exist (A∪B)\(A∩B) =(A∪C)\(A∩C) ⇐⇒ B=C "
    Thanks for help.
    Last edited by MathPro17; Mar 14th 2018 at 10:15 AM.
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    Re: I would like to know for example how I prove or disprove this sentence ?

    Quote Originally Posted by Plato View Post
    In other words, you have now changed to $\large\color{red}{C\setminus A=C\setminus B}$ from $\large\color{blue}{A\setminus C=B\setminus C}$ with the same conclusion?

    If so consider: $A=\{p,q,x,w,y\}~,~B=\{r,s,x,w,y\},~\&~C=\{u,v,x,w ,y\}$ Can $A\subseteq B~?$
    Yes you right , becuase on the page exercises they have written that , that sentence not true and I give them contradict example.

    I have another difficult question that I don't know how to solve :
    "For any A,B,C Sets if exist (A∪B)\(A∩B) =(A∪C)\(A∩C) ⇐⇒ B=C "
    Thanks for help.
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    Re: I would like to know for example how I prove or disprove this sentence ?

    $\Rightarrow$: Let $x \in B$. Show that $x \in C$.
    Case 1: $x \in A$. Then, $x \notin (A\cup B) \setminus (A\cap B) = (A\cup C) \setminus (A \cap C)$. Since $x \in A$, this implies $x \in A\cup C$. Since $x \notin (A\cup C) \setminus (A \cap C)$, it must be that $x \in A\cap C \subseteq C$, so $x \in C$.
    Case 2: $x \notin A$. Then $x \in (A\cup B)$, but $x \notin A\cap B$. So, $x \in (A\cup B) \setminus (A\cap B) = (A\cup C) \setminus (A\cap C)$. Since $x \notin A$, you know $x \notin A\cap C \subseteq A$. Therefore, it must be that $x \in C$.
    $\Leftarrow$: Let $x \in C$. Show that $x \in B$.
    Similarly as above.
    Last edited by SlipEternal; Mar 14th 2018 at 10:42 AM.
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    Re: I would like to know for example how I prove or disprove this sentence ?

    Quote Originally Posted by SlipEternal View Post
    $\Rightarrow$: Let $x \in B$. Show that $x \in C$.
    Case 1: $x \in A$. Then, $x \notin (A\cup B) \setminus (A\cap B) = (A\cup C) \setminus (A \cap C)$. Since $x \in A$, this implies $x \in A\cup C$. Since $x \notin (A\cup C) \setminus (A \cap C)$, it must be that $x \in A\cap C \subseteq C$, so $x \in C$.
    Case 2: $x \notin A$. Then $x \in (A\cup B)$, but $x \notin A\cap B$. So, $x \in (A\cup B) \setminus (A\cap B) = (A\cup C) \setminus (A\cap C)$. Since $x \notin A$, you know $x \notin A\cap C \subseteq A$. Therefore, it must be that $x \in C$.
    $\Leftarrow$: Let $x \in C$. Show that $x \in B$.
    Similarly as above.
    Thanks very much ,
    in the original question I have A "Triangle symbol" B = A "Traingle Symbol" C , I can write this like I have written as : "(A∪B)\(A∩B) =(A∪C)\(A∩C)" ?
    (I havn't write the traingle symbol that is symetric difference because I don't know how to write this in this box if you could explain me I will be gratefull )
    I have to rewrite the question with this : "(A∪B)\(A∩B) =(A∪C)\(A∩C)" to solve it like you do ?
    Last edited by MathPro17; Mar 14th 2018 at 11:23 AM.
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  15. #15
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    Re: I would like to know for example how I prove or disprove this sentence ?

    $A \Delta B = A \Delta C$

    You don't have to solve it the way I showed. There are many different ways to solve these problems.
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