Podzbiorem

Podzbiorem is the instrumental form of the Polish term podzbiór, used in mathematics to refer to the concept of a subset. In set theory, a set A is a subset of a set B if every element of A is also an element of B. Notation: A ⊆ B denotes a (not necessarily proper) subset, while A ⊂ B is commonly used to indicate a proper subset, where A ≠ B. The empty set ∅ is a subset of every set, and every set is a subset of itself (reflexivity).

Examples help illustrate the idea. Let A = {1, 3, 5} and B = {1, 2, 3, 4, 5}. Then A ⊆ B, and A ⊂ B because B contains elements not in A. The power set P(S) is the set of all subsets of S; if |S| = n, then |P(S)| = 2^n. Subset relations are transitive: if A ⊆ B and B ⊆ C, then A ⊆ C.

Subsets support basic operations. If A and B are subsets of a universal set U, then A ∩ B and A ∪ B are also subsets of U. The complement of A with respect to U, written U \ A, is another subset of U. Subsets underpin many mathematical constructions, including functions (as sets of ordered pairs), logic, and combinatorics.

In Polish mathematical usage, podzbiór (and its instrumental form podzbiorem) is used similarly to denote a subset of a given set. The term is fundamental to set theory and underlies definitions of membership, equality of sets, and cardinality.