To prove that sets $A$ and $B$ are equal given the conditions $A \cup C = B \cup C$ and $A \cap C = B \cap C$, one can utilize the properties of set operations. The proof involves demonstrating that any element in $A$ must also be in $B$ and vice versa, leveraging the union and intersection properties. This leads to the conclusion that $A = B$ under the specified conditions.
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Albert said:Three sets $A,B,C$ given:
(1)$A\bigcup C=B\bigcup C$ and (2)$A\bigcap C=B \bigcap C$
Prove: $A=B$
Thanks! very good !Fernando Revilla said: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 $B\subset A.$