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Finite set. In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the ...
In mathematics, a set is a collection of different [ 1] things; [ 2][ 3][ 4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. [ 5] A set may have a finite number of elements or be an ...
Hereditarily finite set. In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself is finite, and all of its elements are finite sets, recursively all the way down to the empty set .
This "finer-than" relation on the set of partitions of X is a partial order (so the notation "≤" is appropriate). Each set of elements has a least upper bound (their "join") and a greatest lower bound (their "meet"), so that it forms a lattice, and more specifically (for partitions of a finite set) it is a geometric and supersolvable lattice.
A block code can also be described as a family of sets, by describing each codeword as the set of positions at which it contains a 1. A topological space consists of a pair. ( X , τ ) {\displaystyle (X,\tau )} where. X {\displaystyle X} is a set (whose elements are called points) and. τ {\displaystyle \tau } is a topology on.
Let be a set and a nonempty family of subsets of ; that is, is a 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.
Definition as von Neumann ordinals. In Zermelo–Fraenkel (ZF) set theory, the natural numbers are defined recursively by letting 0 = {} be the empty set and n + 1 (the successor function) = n ∪ {n} for each n. In this way n = {0, 1, …, n − 1} for each natural number n. This definition has the property that n is a set with n elements.
The set of points P is a finite set. In any finite projective space, each line contains the same number of points and the order of the space is defined as one less than this common number. A subspace of the projective space is a subset X , such that any line containing two points of X is a subset of X (that is, completely contained in X ).