Proving Set Equality

Real Analysis · Axiom Academy

A complete element-chasing proof of De Morgan's Law for sets. Prove the set identity (De Morgan's Law) for arbitrary sets A, B inside a universe U . We use the element-chasing method: show each set is a subset of the other. Visual Check with Venn Diagrams The proof is airtight on its own. As a sanity check, here is each side shaded inside a universe U — the region outside both circles: Identical shaded regions — exactly what the proof established. You've completed a rigorous proof of De Morgan's Law by element-chasing. The reusable ideas: Equality by double inclusion: to prove S = T , show and . Element-chasing: take an arbitrary element of one set and follow its membership through the definitions until it lands in the other. Definitions are the engine: complement ( ), union, and intersection turn set statements into logic. Sets mirror logic: is the set-theoretic face of . Pictures support, proofs decide: the Venn diagram builds intuition, but the two-direction argument is what makes it certain. The same method proves the companion law — try it: start from and chase the element through.

This is the written version of the interactive lesson above. See the full Real Analysis course.