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)
andB.equals(A)
will both return trueA.retainAll(B)
andB.retainAll(A)
will both return false, as the calling operation will not modify the set.A.removeAll(B)
andB.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?