TITLE: Looking for equivalence law names
QUESTION [0 upvotes]: This subject is very foreign to me and has me kind of confused.  This problem seems very easy, but because I'm new to this, its not to me.  I'm not sure of the name of the law I'm using, or if its correct at all.  I'm supposed to prove that:
A \ (A ∩ B) = A \ B

x ∈ A\(A∩B) ⇔ (x ∈ A\A) ∩ (x ∈ A\B))               (Using distribution law)
            ⇔ (x ∈ A\B)                            (which law?)

Does A\A cancel out?  Is that a valid operation?  I can't find it in my notes at all, but I assume the difference between A and A is nothing, or maybe an empty set?  Is (x ∈ A\A) some sort of contradiction?  As it seems to be stating x belongs to set A but not to set A.  Any help clarifying this would be great.  If it does have a law name, the specific name would be helpful as well, as I need to specify what I'm doing to get the to answer.  Thank you.

REPLY [1 votes]: Recall that $A \setminus B = A \cap B^C$. Now,
\begin{align*}
A \setminus (A \cap B) &= A \cap (A \cap B)^C \\
&= A \cap (A^C \cup B^C) \\
&= (A \cap A^C) \cup (A \cap B^C) \\
&= \emptyset \cup (A \setminus B) \\
&= A \setminus B
\end{align*}
To answer one point you had, $A \setminus A = \emptyset$ which can be seen intuitively: Take a set, remove everything, you're left with nothing.
There really isn't a name for this, it's just another way to write relative complements.

REPLY [1 votes]: You are looking at a set difference:
\begin{align}
A\setminus(A\cap B)&\equiv A\cap(A\cap B)^c\tag{by definition}\\[1em]
&\equiv A\cap(A^c\cup B^c)\tag{De Morgan}\\[1em]
&\equiv (A\cap A^c)\cup(A\cap B^c)\tag{why?}\\[1em]
&\equiv A\cap B^c\tag{why?}\\[1em]
&\equiv A\setminus B.\tag{by definition}
\end{align}
If you are being asked to prove an identity about set differences, then you must obviously first know what a set difference is. If you have questions about the margin comments above, then I recommend reading your text more closely or talking to your teacher.
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