1. Set question #1

Why $( A \subseteq B ) \equiv [ A \cap B = A ] \equiv [ A \cup B = B ]$ ?
Why $A \cap ( B - C ) = ( A \cap B) - ( A \cap C)$ ?
Why $A - B = A \oplus ( A \cap B )$

2. Originally Posted by Researcher
Why $( A \subseteq B ) \equiv [ A \cap B = A ] \equiv [ A \cup B = B ]$ ?
Why $A \cap ( B - C ) = ( A \cap B) - ( A \cap C)$ ?
Why $A - B = A \oplus ( A \cap B )$
These are so basic, what is your problem? Do you know basic definitions?
What does $A \subseteq B$ mean to you?
What does $A \cap B$ mean to you?
What does $A \cup B$ mean to you?

3. Originally Posted by Plato
These are so basic, what is your problem? Do you know basic definitions?
What does $A \subseteq B$ mean to you?
What does $A \cap B$ mean to you?
What does $A \cup B$ mean to you?
I know the meanings of them.
I know they seem true intuitively, but i want their proofs!!
I can prove the 3rd. But in proving the 2nd i get a contradiction!
Now can you help me?
If it's necessary, i can show the contradiction.

4. Originally Posted by Researcher
But in proving the 2nd i get a contradiction!
$x \in A \cap B \Leftrightarrow \left[ {\left( {x \in A} \right) \wedge \left( {x \in B} \right)} \right]$

6. Originally Posted by Researcher