1. ## Proof with sets

I have to finish this proof.

Let $S$ and $T$ be sets.

Let $S\oplus T$ be the symmetric difference of those sets.

THM: $\forall ~\mbox{sets} ~ A,B,C, ~\mbox{if}~ A \oplus C = B\oplus C,~\mbox{then}~A=B$.

Prove the above theorem by filling in the missing _____.

Proof(Direct)

Assume 1.) _______
Show 2.) _______

$(\subseteq)$ Let $x \in A$. Show $x\in B$.

Case 1:

3.) ___________________________________

$x \in B$

Case 2: $x \in C$

4.) ___________________________________

$x \in B$

$(\subseteq)$ (<-- should be backwards, but not sure how to do it in latex).

Similarly

Q.E.D

So 1 and 2 are easy, it's 3 and 4 that are rough.

1.) $A \oplus C = B\oplus C$

2.) $A=B$

2. Originally Posted by alikation
I have to finish this proof.

Let $S$ and $T$ be sets.

Let $S\oplus T$ be the symmetric difference of those sets.

THM: $\forall ~\mbox{sets} ~ A,B,C, ~\mbox{if}~ A \oplus C = B\oplus C,~\mbox{then}~A=B$.

Prove the above theorem by filling in the missing _____.

Proof(Direct)

Assume 1.) _______
Show 2.) _______

$(\subseteq)$ Let $x \in A$. Show $x\in B$.

Case 1: ${\color{red}x \not\in C}$

3.) ___________________________________

${\color{red}x \not\in C \Rightarrow x \in A - C \subset A \oplus C = B \oplus C \Rightarrow x \in B \oplus C }$

But since ${\color{red}x \not\in C \Rightarrow x \not\in C-B \Rightarrow x \in B-C}$

$x \in B$

Case 2: $x \in C$

4.) ___________________________________

${\color{red}x \not\in A \oplus C \Rightarrow x \not\in B \oplus C \Rightarrow x \in B \cap C \Rightarrow x \in B}$

$x \in B$

$(\subseteq)$ (<-- should be backwards, but not sure how to do it in latex).

Similarly

Q.E.D

So 1 and 2 are easy, it's 3 and 4 that are rough.

1.) $A \oplus C = B\oplus C$

2.) $A=B$
..