Set theory is a branch of mathematics that deals with the study of collections of objects, called sets. It provides a foundation for many other areas of mathematics, such as algebra, calculus, and logic.
In set theory, a subset is a collection of elements from a larger set. A subset is considered to be a part or a smaller version of the larger set, and it may contain some or all of the elements of the larger set.
A proper subset is a subset that contains some but not all of the elements of a larger set. An improper subset is a subset that contains all of the elements of a larger set. In other words, an improper subset is a subset that is equal to the larger set.
A subset is expressed using the notation A ⊆ B, which means that the set A is a subset of the set B. This notation is read as "A is a subset of B" or "A is contained in B".
The cardinality of a subset is the number of elements it contains. It is denoted by |A|, where A is the subset. For example, if a set A = {1, 2, 3, 4}, then |A| = 4, as it contains four elements.