Proving $A=B$ from Sets $A,B,C$

[Albert1]

Three sets $A,B,C$ given:

(1)$A\bigcup C=B\bigcup C$

and

(2)$A\bigcap C=B \bigcap C$

Prove: $A=B$

 

[Fernando Revilla]

Three sets $A,B,C$ given:

(1)$A\bigcup C=B\bigcup C$ and (2)$A\bigcap C=B \bigcap C$
Prove: $A=B$

Spoiler

Let's prove that $A\subset B$ $$x\in A\Rightarrow x\in A\cup C=B\cup C\Rightarrow \left \{ \begin{matrix} x\in B\\\vee\\x\in C\end{matrix}\right.$$ $$\Rightarrow \left \{ \begin{matrix} x\in B\\\vee\\x\in A\cap C=B\cap C\end{matrix}\right.\Rightarrow \left \{ \begin{matrix} x\in B\\\vee\\x\in B\end{matrix}\right.\Rightarrow x\in B.$$ In the same way, we could prove that $B\subset A.$

 

[Albert1]

Spoiler

Let's prove that $A\subset B$ $$x\in A\Rightarrow x\in A\cup C=B\cup C\Rightarrow \left \{ \begin{matrix} x\in B\\\vee\\x\in C\end{matrix}\right.$$ $$\Rightarrow \left \{ \begin{matrix} x\in B\\\vee\\x\in A\cap C=B\cap C\end{matrix}\right.\Rightarrow \left \{ \begin{matrix} x\in B\\\vee\\x\in B\end{matrix}\right.\Rightarrow x\in B.$$ In the same way, we could prove that $B\subset A.$

Thanks! very good !
sol-1

Spoiler

$A=A\cup(A\cap C)
=A\cup(B\cap C)
=(A\cup B)\cap (A\cup C)$
$=(B\cup A)\cap (B\cup C)
=B\cup (A\cap C)$
$=B\cup(B\cap C)=B$

sol-2

Spoiler

$A=A\cap(A\cup C)
=A\cap(B\cup C)
=(A\cap B)\cup (A\cap C)$
$=(B\cap A)\cup (B\cap C)
=B\cap (A\cup C)$
$=B\cap(B\cup C)=B$