A *set* is an object defined by its elements. It satisfies certain other properties in different axiomatic systems: 1. [[Naive Set Theory]]. 2. [[Zermelo Frankel set theory (ZFC)]]. # Properties **Set operations and algebraic properties**: We can define the [[Set Union|union, intersection & difference]] of sets. Also we define [[Cartesian Product|Cartesian product]] of two sets as the set of ordered pairs. **Relations between set elements:** A [[Binary Relation|binary relation]] between the elements of two sets is modelled as a subset of their cartesian product. A [[Function|function]] is a binary relation between two sets that maps each element in the first to a unique element in the second. **Relations between sets:** (Subset) A set it said to be a [[Subset|subset]] of another if all of its elements are also elements of the other. The set of a all of subset of a given set is known as its [[Power Set Axiom|power set]]. (Size) Two sets are said to have the same [[Cardinality|cardinality]] if there exists a bijection between them. [[Cantor's Theorem|Cantor's theorem]] gives that the cardinality of a given set's power set is strictly greater than the given set. Also note that [[Schroeder-Bernstein Theorem|Schroeder-Bernstein theorem]] implies, informally, that if two cardinalities are both less than or equal to each other, then they are equal. # Applications **Multiset**: A [[Multiset]], informally, is a collection objects up to multiplicity. **Numbers**: The [[Natural Numbers|natural numbers]] are constructed inductively by the axiom of infinity. We can use them to construct other numbers: [[Integers|integers]]; [[Rational Number|rationals]] & [[Real Numbers|reals]]. **Algebra**: An [[Algebraic Structure|algebraic structure]] is a set together with operations that satisfy a finite set of rules.