Conjunto

Conjunto is the term used in Spanish and Portuguese for the mathematical object known in English as a set. In mathematics, a conjunto is a collection of distinct objects called elements, considered as a single entity. The elements either belong to the conjunto or they do not; the order of elements and repetitions are irrelevant. Conjuntos can be finite or infinite. Notation uses curly braces, as in {a, b, c}, to denote a conjunto with elements a, b, and c. The empty set, written ∅ or {}, is the unique conjunto with no elements. For an element x, x ∈ A means x is a member of the conjunto A, and x ∉ A means x is not a member.

Subsets and equality: A conjunto A is a subset of B if every element of A is also an element of B, written A ⊆ B. If A ⊆ B and A ≠ B, then A is a proper subset of B, written A ⊂ B. Two conjuntos are equal when they have exactly the same elements, A = B.

Operations: Union and intersection are central: A ∪ B contains elements in A or B (or both); A ∩ B contains elements common to both. Difference, written A \ B or A − B, contains elements in A but not in B. The complement of A relative to a universal set U is U \ A. The Cartesian product A × B is the conjunto of all ordered pairs (a, b) with a ∈ A and b ∈ B.

Cardinality and constructions: The size of a finite conjunto A is |A| = n. Infinite conjuntos have infinite cardinalities, denoted by symbols such as ℵ0. The power set P(A) is the conjunto of all subsets of A and has cardinality 2^|A| for finite A. Conjuntos form the foundation of set theory, the branch of mathematical logic that studies their properties under axioms like Zermelo–Fraenkel. They underpin many areas of mathematics and its formal language.