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

1. ## 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.

2. ## Re: I would like to know for example how I prove or disprove this sentence ? Originally Posted by MathPro17 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 $B~?$

3. ## Re: I would like to know for example how I prove or disprove this sentence ? Originally Posted by Plato 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.

4. ## 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.

5. ## Re: I would like to know for example how I prove or disprove this sentence ? Originally Posted by MathPro17 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*}

6. ## 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 ?

7. ## Re: I would like to know for example how I prove or disprove this sentence ? Originally Posted by MathPro17 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 $A=B \Longrightarrow (A\subseteq B \wedge B\subseteq A)$

8. ## Re: I would like to know for example how I prove or disprove this sentence ? Originally Posted by SlipEternal 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

9. ## Re: I would like to know for example how I prove or disprove this sentence ? Originally Posted by MathPro17 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 $x \in B$.

10. ## Re: I would like to know for example how I prove or disprove this sentence ? Originally Posted by MathPro17 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 $A\subseteq B~?$

11. ## Re: I would like to know for example how I prove or disprove this sentence ? Originally Posted by SlipEternal 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.

12. ## Re: I would like to know for example how I prove or disprove this sentence ? Originally Posted by Plato 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.

13. ## 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.

14. ## Re: I would like to know for example how I prove or disprove this sentence ? Originally Posted by SlipEternal $\Rightarrow$: 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 ?

15. ## 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.