Set theory

• How Do You Find the Cardinality of a Set Containing Sets?
  To find the cardinality of a set containing sets, count its distinct outer elements; do not add the numbers of elements inside those sets.
• What Is the Cartesian Product with the Empty Set?
  The Cartesian product of any set with the empty set is empty: neither A × ∅ nor ∅ × A contains an ordered pair.
• Can a Set Be an Element of Itself?
  In ZF and ZFC, the Axiom of Foundation rules out x ∈ x. See the short proof, why the axiom matters, and how non-well-founded theories differ.