Chapter 1 — Sets

A set is a well-defined collection of objects, fundamental to modern mathematics. Georg Cantor developed set theory in the 1870s while studying trigonometric series.

Sets form the foundation of modern mathematics and are used across geometry, sequences, probability, and other branches. A set is a well-defined collection where you can definitively determine membership. Sets are represented in roster form (listing elements) or set-builder form (defining properties). Key concepts include finite/infinite sets, subsets, unions, intersections, complements, and operations following laws like commutativity and distributivity. The empty set contains no elements, while the universal set provides context for all operations.

1. 01A set is a well-defined collection of objects; membership is always decidable
2. 02Two representation methods: roster form {1,2,3} and set-builder form {x: condition}
3. 03Empty set (φ) contains no elements; finite sets have definite cardinality; infinite sets have unlimited elements
4. 04A subset (A ⊂ B) means every element of A is also in B; proper subsets exclude equality
5. 05Set operations include union (∪), intersection (∩), difference (-), and complement ('); follow commutativity, associativity, and distributivity laws
6. 06De Morgan's laws: (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'