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These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one's initial intuition.
For example, the intersection of the x-axis and y-axis in under addition is the trivial subgroup. More generally, the intersection of an arbitrary collection of subgroups of G is a subgroup of G. The union of subgroups A and B is a subgroup if and only if A ⊆ B or B ⊆ A.
A partition of a set X is a set of non-empty subsets of X such that every element x in X is in exactly one of these subsets [2] (i.e., the subsets are nonempty mutually disjoint sets). Equivalently, a family of sets P is a partition of X if and only if all of the following conditions hold: [3]
(An example of this is the subset {: <} of . It has upper bounds, such as 1.5, but no supremum in Q {\displaystyle \mathbb {Q} } .) Consequently, partially ordered sets for which certain infima are known to exist become especially interesting.
In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: 1 − 2 is not a natural number, although both 1 and 2 are.
For example, the natural numbers with their standard order. A chain is a subset of a poset that is a totally ordered set. For example, {{}, {}, {,,}} is a chain. An antichain is a subset of a poset in which no two distinct elements are
Nested set representing a biological taxonomy example. Outside-in: order, family, genus, species. A nested set collection or nested set family is a collection of sets that consists of chains of subsets forming a hierarchical structure, like Russian dolls .
Formally, let = (,) be any graph, and let be any subset of vertices of G. Then the induced subgraph G [ S ] {\displaystyle G[S]} is the graph whose vertex set is S {\displaystyle S} and whose edge set consists of all of the edges in E {\displaystyle E} that have both endpoints in S {\displaystyle S} . [ 1 ]