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For instance, had been declared as a subset of , with the sets and not necessarily related to each other in any way, then would likely mean instead of . If it is needed then unless indicated otherwise, it should be assumed that X {\displaystyle X} denotes the universe set , which means that all sets that are used in the formula are subsets of X ...
A set A of natural numbers is an IP set if there exists an infinite set D such that FS(D) is a subset of A. Equivalently, one may require that A contains all finite sums FS((n i)) of a sequence (n i). Some authors give a slightly different definition of IP sets: They require that FS(D) equal A instead of just being a subset.
In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment).
A partition of a set is defined as a family of nonempty, pairwise disjoint subsets of whose union is . For example, B 3 = 5 {\displaystyle B_{3}=5} because the 3-element set { a , b , c } {\displaystyle \{a,b,c\}} can be partitioned in 5 distinct ways:
Let be a set and a nonempty family of subsets of ; that is, is a nonempty subset of the power set of . Then is said to have the finite intersection property if every nonempty finite subfamily has nonempty intersection; it is said to have the strong finite intersection property if that intersection is always infinite.
Two subsets and are separated precisely when they are disjoint and each is disjoint from the other's derived set ′ = = ′. [6] A bijection between two topological spaces is a homeomorphism if and only if the derived set of the image (in the second space) of any subset of the first space is the image of the derived set of that subset.
The 5th roots of unity in the complex plane form a group under multiplication. Each non-identity element generates the group. In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses.
A subset S is both saturated and multiplicatively closed if and only if S is the complement of a union of prime ideals. [4] In particular, the complement of a prime ideal is both saturated and multiplicatively closed. The intersection of a family of multiplicative sets is a multiplicative set. The intersection of a family of saturated sets is ...