zbiory

Zbiory, or sets, are a fundamental concept in mathematics. A zbiory is a well-defined collection of objects, called elements, such that membership is unambiguous: for any object x, x is either in the set or it is not. Sets may be finite or infinite, and are typically denoted by uppercase letters such as A, B, or S.

Basic notation includes the symbol ∈, where x ∈ A means x is an element of A, and ∉ for not belonging. The empty set ∅ contains no elements, and the universal set U contains all objects under consideration within a given context. A subset is a set whose elements are all contained in another set: A ⊆ B. If A ⊆ B and A ≠ B, then A is a proper subset of B, often written as A ⊂ B.

Common operations on sets include union (A ∪ B), which combines elements from both sets; intersection (A ∩ B), which contains elements common to both; difference (A \ B), which contains elements in A but not in B; and complement relative to a universal set (A^c). Sets can be related through membership, relations, and functions, and the Cartesian product A × B forms the set of all ordered pairs (a, b) with a ∈ A and b ∈ B. The power set P(A) is the set of all subsets of A.

Finite sets have a finite cardinality |A|, while infinite sets can be countable or uncountable. Classic examples include the set of natural numbers N, integers Z, and rationals Q (countable), versus the real numbers R (uncountable). Zbiory provide a formal basis for almost all areas of mathematics and related disciplines.