mnoiny

Mnoziny, the Czech term for mathematical sets, are a fundamental concept in mathematics. A set is a collection of distinct objects called elements. The order of elements does not matter, and an element may appear at most once in a set.

Sets are typically denoted by curly braces, for example A = {1, 2, 3}. The symbol ∈ means “is an element of”; thus 2 ∈ A holds, while 4 ∉ A does not.

Important notions include subsets, unions, intersections and complements. A ⊆ B means every element of A is also in B; if A ⊆ B and B has elements not in A, A is a proper subset of B. The union A ∪ B contains all elements that are in A or B; the intersection A ∩ B contains elements common to both. The difference A \ B contains elements in A but not in B. The complement of A, relative to a universal set U, consists of elements of U not in A.

Sets can be defined by properties using set-builder notation, such as A = {x ∈ ℝ | x^2 < 2}. The cardinality |A| counts the elements; finite sets have finite cardinality, the empty set ∅ has cardinality 0, and infinite sets have infinite cardinalities. The power set P(A) is the set of all subsets of A, with |P(A)| = 2^|A|. The Cartesian product A × B is the set of ordered pairs (a, b).

Common symbol systems include ℕ, ℤ, ℚ, and ℝ representing natural numbers, integers, rationals, and real numbers. Sets underpin much of mathematics and are used in logic, computer science, and data theory.