Set Operations and Identities

Intro to Proofs · Axiom Academy

LESSON Set Operations and Identities Master union, intersection, complement, and difference — then prove the identities that govern them, including De Morgan's laws. Let A and B be sets drawn inside a universe U . Four operations build every other set expression. The animation shades each region in turn — watch which part of the Venn diagram each one selects. Elements in A or B (or both) — everything inside either circle. Elements in both A and B — only the overlapping lens. Elements not in A — everything in U outside circle A. Elements in A but not in B — the part of A clear of B. To prove two sets are equal, show that an arbitrary element x belongs to the left-hand side if and only if it belongs to the right-hand side. The animation follows one element x as the region fills in and lands exactly on A . Instead of tracking individual elements, manipulate the set expression with known identities — just like simplifying an algebraic equation. The animation shades the same identity region-by-region: first , then , and the union settles on all of A . [Distributive] [Complement Law] [Identity Law] De Morgan's laws relate the complement of a union to the intersection of complements (and vice versa). The animation shades on the left and on the right — watch the two regions turn out to be identical . The complement of a union is the intersection of complements. The complement of an intersection is the union of complements. 5. Proving De Morgan's First Law

This is the written version of the interactive lesson above. See the full Intro to Proofs course.