Subset
A set whose elements all belong to another set.
In mathematics, a subset is a set whose elements are all contained within another set. The concept of subset, denoted by A ⊆ B, is fundamental to set theory, establishing a relationship of inclusion between sets. A proper subset, denoted by A ⊊ B, occurs when the subset is not equal to the superset, meaning the superset contains at least one element not in the subset.
- definition
- A is a subset of B if every element of A is also an element of B
- notation
- A ⊆ B (subset), A ⊊ B (proper subset)
- key_properties
- Reflexivity, transitivity, antisymmetry
- empty_set
- The empty set is a subset of every set
- power_set
- The set of all subsets of a set S is its power set, denoted P(S)
Lore & Background
The subset relation is defined by the condition that for all elements x, if x is in A then x is in B. This can be proven using the element argument, where one takes an arbitrary element of A and shows it is also an element of B. The empty set, having no elements, is vacuously a subset of any set. The inclusion relation ⊆ is a partial order on the power set of any set S.
Reader's Guide
The subset concept is central to set theory and mathematics. It allows for the comparison of sets and the definition of equality via antisymmetry: if A ⊆ B and B ⊆ A, then A = B. The distinction between subset and proper subset is important, though notation varies among authors—some use ⊂ to mean subset (reflexive) and others to mean proper subset (irreflexive). The empty set's property of being a subset of every set is a foundational logical truth. The power set, the set of all subsets, illustrates how the subset relation structures mathematical hierarchies. Examples include finite sets like {1, 2} being a proper subset of {1, 2, 3}, and infinite sets like the natural numbers being a proper subset of the rational numbers, sometimes with equal cardinality.
Did You Know?
- The empty set is a subset of every set.
- A set is always a subset of itself (reflexivity).
- If A is a proper subset of B, then B is not a proper subset of A (asymmetry).
- Some authors use ⊂ to mean subset (reflexive) while others use it to mean proper subset (irreflexive).
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