Subsets

A subset is a fundamental concept in set theory, referring to a set whose elements are all contained within another set. If A and B are sets, then A is a subset of B (denoted as A ⊆ B) if every element of A is also an element of B. Conversely, if some elements of A are not in B, then A is not a subset of B.

A special case is the empty set, which contains no elements. The empty set is considered a subset of every set, as the condition that all elements of the empty set are in another set holds trivially. A set that contains exactly the same elements as another set is called an equal set. If all elements of A are in B but B contains additional elements, then A is a proper subset of B, denoted as A ⊂ B.

Subsets are crucial in understanding the structure and relationships between different sets. They are used extensively in mathematics, logic, computer science, and related fields to form the basis for functions, relations, and various types of algebraic structures.

The power set of a given set is the set of all its subsets, including the set itself and the empty set. The concept of subsets facilitates various mathematical operations, proofs, and the study of set hierarchies, making it a foundational element in the discipline of set theory.