Set Operations and Properties

Real Analysis · Axiom Academy

Union, intersection, and complement build new sets from old — and complement turns out to be a mirror that swaps the other two. Picture two sets A and B as overlapping circles inside a universe U (the surrounding box). Each operation keeps a different region. Watch the shaded region move as we run through all three. Complement — in U but not in A Set operations obey an algebra much like arithmetic — they're commutative, associative, and distributive (the reference cards below). The deepest law concerns how complement interacts with union. Watch the left panel shade the complement of the union, the right panel shade the intersection of the two complements — they land on the same region. The supporting algebra (verified, no animation needed): 3. The Second Law — and the Duality The first law has a perfect twin. Run the same comparison, but start from the intersection : the complement of on the left equals the union of the two complements on the right. The operations have simply swapped places. The big picture — duality. Put the two laws side by side and a symmetry appears: complement acts like a mirror that swaps with . Flip any true set identity by exchanging every and every set with its complement, and you get another true identity for free. — complement turns union into intersection. — and intersection into union. Same rule, mirrored.

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