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Albert1
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Three sets $A,B,C$ given:
(1)$A\bigcup C=B\bigcup C$
and
(2)$A\bigcap C=B \bigcap C$
Prove: $A=B$
(1)$A\bigcup C=B\bigcup C$
and
(2)$A\bigcap C=B \bigcap C$
Prove: $A=B$
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 $B\subset A.$
To prove that two sets A and B are equal, you need to show that every element in set A is also in set B, and vice versa. This can be done by using the subset method, where you show that A is a subset of B and B is a subset of A. Alternatively, you can use the element method, where you show that every element in A is also in B and every element in B is also in A.
Proving that two sets are equal is important because it allows us to show that they have the same elements, and therefore, they can be used interchangeably. This is especially useful in mathematics, where we often need to manipulate sets and perform operations on them.
Yes, two sets can still be proven to be equal even if they are in different forms. This is because the form of a set does not affect its elements. As long as you can show that all the elements in one set are also in the other, the sets can be considered equal.
The reflexive property states that every set is equal to itself. When we prove that two sets are equal, we are essentially showing that they are reflexive, or that they are equal to themselves. This is because we are showing that every element in one set is also in the other, which means that both sets are equal to themselves.
Yes, the transitive property can be used to prove that two sets are equal. This property states that if A is equal to B and B is equal to C, then A is also equal to C. Therefore, if you can prove that set A is equal to set B and set B is equal to set C, then you can conclude that set A is also equal to set C.