Subset

A set is a subset of another set when every element of the first is also an element of the second. A set is a subset of itself, and the empty set is a subset of every set.

Set: A collection of distinct objects, treated as a single mathematical object. Subsethood compares the elements contained in two sets.
Transitive relation: A relation is transitive when one step from a first object to a second and another to a third imply a direct relation to the third. If A is a subset of B and B of C, every element of A belongs to C.
Proper subset: A set contained in another set but not equal to it. It strengthens subsethood by requiring at least one element of the larger set to be absent from the smaller.
Partially ordered set: A set equipped with a reflexive, antisymmetric, and transitive relation. The subsets of any set form a partially ordered set under inclusion.
Set membership: The relation stating that an object is an element of a set. A subset relation holds exactly when each membership in one set also holds in the other.
Intersection: The set of elements shared by two or more sets. An intersection is a subset of each set being intersected.
Set equality: Equality of sets, determined by their having exactly the same elements. Two sets are equal precisely when each is a subset of the other.
Boolean algebra: An algebraic structure with operations corresponding to logical conjunction, disjunction, and negation. Subsets of a fixed set form a Boolean algebra under intersection, union, and complement.
Empty set: The unique set containing no elements. It is a subset of every set because it has no element that could violate the condition.
Union: The set containing every element that belongs to at least one of several sets. Each set in a union is a subset of the resulting union.

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