Proving A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Intro to Proofs · Axiom Academy

EXAMPLE Proving the Distributive Law for Sets A complete element-chasing proof that . Prove the distributive law for sets: . We will use the element-chasing method — pick an arbitrary element, follow it through the definitions, and show each side is a subset of the other. Three sets A , B , C inside a universe U . The proof tracks a single element x — no shading needed. Key Takeaways from Element-Chasing Proofs Nice work — you completed a rigorous proof of the distributive law using the element-chasing method. Here is what makes the technique powerful: Set equality = two inclusions: to show A = B , prove and . This is the fundamental approach for every set-equality proof. Start from the definitions: unpack and into their logical meaning — "and" for intersection, "or" for union. Use case analysis on "OR": when a disjunction appears in your hypothesis, prove the conclusion in each case separately so the argument is complete. Track one element: follow a single arbitrary x all the way through. Showing its membership on one side forces membership on the other is exactly what an inclusion means. Both directions are required: proving only is half a proof. The reverse inclusion mirrors it — same definitions, same case split. This element-chasing technique generalizes: it proves De Morgan's laws, associativity, and many other set identities. Master it and you have a reliable tool for rigorous reasoning about sets.

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