Sets

Sets are collections of distinct objects, called elements. The elements can be numbers, letters, or more abstract objects. A set is denoted by uppercase letters, and its elements are listed inside braces, for example {1, 2, 3}. The empty set, written as ∅, has no elements. The order of elements and repetition do not matter.

Two sets are equal if they contain exactly the same elements. A is a subset of B, written A ⊆ B, if every element of A is also an element of B. If B has an element not in A, then A is a proper subset of B, written A ⊂ B. Set-builder notation describes sets by a rule: {x | condition}.

Key operations include:

- Union: A ∪ B contains elements in A or B.

- Intersection: A ∩ B contains elements in both.

- Difference: A \ B contains elements in A but not in B.

- Complement: the complement of A, relative to a universal set U, consists of elements in U not in A.

A derived notion is the power set P(A), the set of all subsets of A. For a finite set with n elements, |P(A)| = 2^n. The cardinality |A| measures size; sets can be finite or infinite. Infinite sets may be countably infinite (e.g., natural numbers ℕ, integers ℤ, rationals ℚ) or uncountable (e.g., real numbers ℝ).

Sets provide foundations for many mathematical concepts, including functions as sets of ordered pairs, relations, and Cartesian products. Venn diagrams help visualize relationships between a few sets. In formal mathematics, set theory is developed axiomatically, with theories such as Zermelo–Fraenkel.