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

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

Originally Posted by SlipEternal
$A \Delta B = A \Delta C$

You don't have to solve it the way I showed. There are many different ways to solve these problems.
I mean if I have this : " AΔB=AΔC" I can rewrite to this on the question like I did : " (A∪B)\(A∩B) =(A∪C)\(A∩C)" without prove it , right?
Thanks.

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

If you have proven that $A \Delta B = (A \cup B) \setminus (A \cap B)$, then yes, you can rewrite the question like you did. The "proof" would be to reference your previous proof that it is true.

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

Originally Posted by SlipEternal
If you have proven that $A \Delta B = (A \cup B) \setminus (A \cap B)$, then yes, you can rewrite the question like you did. The "proof" would be to reference your previous proof that it is true.
But I don't prove that I just rewrite it and try to proven the all sentence , this isn't okay ?
How could I prove this question in this syntax :
" For any A,B,C Sets if exist AΔB=AΔC ⇐⇒ B=C " ?

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

Exactly the same way. Use the definition of the symmetric difference.

$\Rightarrow$: Let $x \in B$.
Case 1: $x \in A \Delta B$. Since $x \in B$, it must be that $x \notin A$. But, we also have $A\Delta B = A\Delta C$, so $x \in A$ or $x\in C$, but not $x \in A\cap C$. Since $x \notin A$, we know $x \in C$.
Case 2: $x \notin A \Delta B$. Since $x \in B$, it must be that $x \in A$. Then, since $A\Delta B = A \Delta C$, we know that $x \notin A \Delta C$. But, we know $x \in A$, so it must be that $x \in C$.
$\Leftarrow$: Similar to above.

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

Originally Posted by SlipEternal
Exactly the same way. Use the definition of the symmetric difference.

$\Rightarrow$: Let $x \in B$.
Case 1: $x \in A \Delta B$. Since $x \in B$, it must be that $x \notin A$. But, we also have $A\Delta B = A\Delta C$, so $x \in A$ or $x\in C$, but not $x \in A\cap C$. Since $x \notin A$, we know $x \in C$.
Case 2: $x \notin A \Delta B$. Since $x \in B$, it must be that $x \in A$. Then, since $A\Delta B = A \Delta C$, we know that $x \notin A \Delta C$. But, we know $x \in A$, so it must be that $x \in C$.
$\Leftarrow$: Similar to above.
Thanks , I will try the next direction and I would be happy if you can correct me :
⇐B=C ;
Let x∈B and from The given information x∈C.
Case 1(We want to show that AΔB ⊆ AΔC): x∈AΔB and we want to show that x∈AΔC, if x∈AΔB then x∉A , we know that x∈C from given info then x∈AΔC.

Case 2(We want to show that AΔC ⊆ AΔB): x∈AΔC , we know that x∈C that mean it must be x∉A . we now that x∈B from info given on top and x∉A then x∈AΔB .
proved .

Is that Okay ?

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

Originally Posted by MathPro17
Thanks , I will try the next direction and I would be happy if you can correct me :
⇐B=C ;
Let x∈B and from The given information x∈C.
Case 1(We want to show that AΔB ⊆ AΔC): x∈AΔB and we want to show that x∈AΔC, if x∈AΔB then x∉A , we know that x∈C from given info then x∈AΔC.

Case 2(We want to show that AΔC ⊆ AΔB): x∈AΔC , we know that x∈C that mean it must be x∉A . we now that x∈B from info given on top and x∉A then x∈AΔB .
proved .

Is that Okay ?

Close, but for $\Leftarrow$, you are assuming $B=C$ to prove that $A\Delta B = A\Delta C$. So, instead of $x \in B$, you would start with $x \in A\Delta B$ and prove that $x \in A \Delta C$.
Also, I did not complete the proof for $\Rightarrow$. I assumed $A\Delta B = A \Delta C$ and proved that $B \subseteq C$. I did not prove that $C\subseteq B$, which is needed for $B=C$. That can be done with a single line saying "The proof that $C \subseteq B$ follows exactly the same as the proof for $B\subseteq C$."

So, here is the proof:
$\Leftarrow:$ Given $B=C$, we want to show $A\Delta B = A \Delta C$ for any set $A$. First, we will show that $A\Delta B \subseteq A \Delta C$. Let $x \in A\Delta B$. We want to show that $x \in A\Delta C$.
Case 1: $x \in A$. Then, by the definition of symmetric difference, $x \notin B$. Since $B=C$, we know $x \notin C$. Therefore, since $x \in A, x\notin C$, we know $x \in A\Delta C$ again by the definition of symmetric difference.
Case 2: $x \notin A$. Then, since $x\in A\Delta B$, it must be that $x \in B$, and since $B=C$, we know $x \in C$. Since $x \notin A, x \in C$, we have $x \in A \Delta C$, again by the definition of symmetric difference.
Since this is true for any arbitrary $x \in A\Delta B$, we have $A\Delta B \subseteq A\Delta C$ for any set $A$.
The proof that $A\Delta C \subseteq A\Delta B$ follows the exact same proof (just swapping $B$ and $C$). There is no reason to rewrite it.

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

You helped me alot , I sorry for the delay becuase I am not be here since then .

8. ## 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 groups exist A\C=B\C And A ∩ C =B ∩ C , then A⊆ B.

Thanks very much for help.
A rather obvious point is that conditions all have "=" so that if your were to reverse A and B they would still be true! But that would imply, if the given statement is true, that you can prove both "$\displaystyle A\subset B$" and "B\subset A".

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