TITLE: Is the statement $A \in A$ true or false?
QUESTION [14 upvotes]: In last week's discrete math homework, one question had us evaluate the truthness of several set notation statements, where $A$ was defined as an arbitrary set.  One such statement was 
$$A \in A$$
I selected true as my answer.  The reasons include

The definition of $A \in B$ means that for all elements in $A$, there must be a one and only one corresponding match in $B$.  Let $A= \{1, 2, 4, 7\}$ and $B= \{1, 2, 4, 7\}$.  Therefore, $A \in B$ identical to $A \in A$, which is shorthand for $\{1, 2, 4, 7\} \in \{1, 2, 4, 7\}$
Set operations in most programming languages are implementations of set theory in computer code and return true if the code evaluates the equality of two sets referencing the same location.  In Java, if we initialize a Set A and B to the values in #1, 

A.equals(B) and B.equals(A) will both return true
A.retainAll(B) and B.retainAll(A) will both return false, as the calling operation will not modify the set.
A.removeAll(B) and B.removeAll(A), if called singularly,  will return true and will result in an empty set


In an email, the professor cited Wikipedia's Axiom of Regularity as a source for discussion on the issue, and it holds that $A\notin A$ due to the definition of disjoint.
However, Van Ormen Quine's New Foundations set theory allows for the existence of a universal set $V$ and that $V\in V$.  Most of that math is beyond my comprehension, but the article cited seems to fit my arguments.
What is the correct answer?

REPLY [3 votes]: What if the question were "does x = 1"? If you're using a system that does not include 1, then it is certainly false. But even if you're in a system which has 1, that doesn't mean that x has to be equal to it.
Analogously, there are some forms of set theory where $A\in A$ is always false, and others in which it is only sometimes false. In either case you should not say that the statement is true, because it isn't known to be true.
In your example you look at A = {1,2,4,7}. For that example $A\in A$ is false, because A does not equal 1, 2, 4, or 7.