partimengden

Partimengden is a fundamental concept in set theory, referring to a subset of a given set. If A and X are sets, A is a partimengden (subset) of X when every element of A is also an element of X. This relation is usually written as A ⊆ X. If A ⊆ X and A ≠ X, A is called a proper subset (often denoted A ⊊ X). The relation is reflexive (A ⊆ A) and, when both A ⊆ B and B ⊆ A hold, A = B.

Examples help illustrate the idea. Let X = {1, 2, 3}. Then ∅ ⊆ X and {1} ⊆ X, as does {1, 2} ⊆ X. The set {1, 2, 3} ⊆ X holds, but it is not a proper subset because it equals X.

Key properties include transitivity: If A ⊆ B and B ⊆ C, then A ⊆ C. Also, if A ⊆ B, then A ∪ B = B and A ∩ B = A. The empty set is a subset of every set, and every set is a subset of itself.

The collection of all partimengden of a given set S is called the power set, denoted P(S). Its size is 2^|S|, reflecting that each element may be either included or excluded from a subset.

Subsets play a central role in proofs, constructions, and probability, where events are often modeled as subsets of a sample space. Notation can vary between texts, so definitions should be checked for whether ⊂ denotes subset or proper subset.