Elementary set theory

• Is (A ∩ B) ∪ C = A ∩ (B ∪ C)?
  The proposed identity is false for arbitrary sets. See a counterexample, the correct distributive law, and the exact condition for equality.
• How Do You List the Elements of a Set?
  A set’s elements can be listed only when its definition is known. Learn how to expand set-builder notation and write finite or infinite sets.
• Is There an Injection Between the Natural and Rational Numbers?
  Yes: inclusion gives an injection from N to Q, and reduced fractions can be encoded as natural-number pairs to inject Q back into N.
• In What Sense Is a Function a Set?
  A function can be represented as a set of input-output ordered pairs; its graph records the mapping, while a triple can also record its codomain.
• Are the Positive Multiples of 17 and 11 in Bijection?
  The positive multiples of 17 and 11 are in bijection: match 17k with 11k for each positive integer k, and verify the map is one-to-one and onto.
• 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.
• What Does It Mean for a Set Not to Contain the Empty Set?
  A set that does not contain the empty set satisfies ∅ ∉ A; this differs from being nonempty and from the fact that ∅ is a subset of every set.