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For example, a complex number can be represented as a 2‑tuple of reals, a quaternion can be represented as a 4‑tuple, an octonion can be represented as an 8‑tuple, and a sedenion can be represented as a 16‑tuple. Although these uses treat ‑uple as the suffix, the original suffix was ‑ple as in "triple" (three-fold) or "decuple" (ten ...
In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B. [1] In terms of set-builder notation, that is [2][3] A table can be created by taking the Cartesian product of a set of rows and a set of columns.
In mathematics, a finitary relation over a sequence of sets X 1, ..., X n is a subset of the Cartesian product X 1 × ... × X n; that is, it is a set of n-tuples (x 1, ..., x n), each being a sequence of elements x i in the corresponding X i. [1] [2] [3] Typically, the relation describes a possible connection between the elements of an n-tuple.
The body is a set of tuples. A tuple is a collection of n values, where n is the relation's degree, and each value in the tuple corresponds to a unique attribute. [6] The number of tuples in this set is the relation's cardinality. [7]: 17–22 Relations are represented by relational variables or relvars, which can be reassigned.
For any natural number n, the set R n consists of all n-tuples of real numbers (R).It is called the "n-dimensional real space" or the "real n-space".An element of R n is thus a n-tuple, and is written (,, …,) where each x i is a real number.
To convert an inversion table d n, d n−1, ..., d 2, d 1 into the corresponding permutation, one can traverse the numbers from d 1 to d n while inserting the elements of S from largest to smallest into an initially empty sequence; at the step using the number d from the inversion table, the element from S inserted into the sequence at the ...
The resulting algebraic object satisfies the axioms for a group. Specifically: Associativity The binary operation on G × H is associative. Identity The direct product has an identity element, namely (1 G, 1 H), where 1 G is the identity element of G and 1 H is the identity element of H.
A basic example of a vector space is the following. For any positive integer n, the set of all n -tuples of elements of F forms an n -dimensional vector space over F sometimes called coordinate space and denoted Fn. [ 1 ] An element of Fn is written. where each xi is an element of F.