For the non-empty product that equals to zero, see zero-product property.
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question), just as the empty sum—the result of adding no numbers—is by convention zero, or the additive identity.[1][2][3][4] When numbers are implied, the empty product becomes one.
The term empty product is most often used in the above sense when discussing arithmetic operations. However, the term is sometimes employed when discussing set-theoretic intersections, categorical products, and products in computer programming.
^Jaroslav Nešetřil, Jiří Matoušek (1998). Invitation to Discrete Mathematics. Oxford University Press. p. 12. ISBN 0-19-850207-9.
^A.E. Ingham and R C Vaughan (1990). The Distribution of Prime Numbers. Cambridge University Press. p. 1. ISBN 0-521-39789-8.
^Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, p. 9, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001
^David M. Bloom (1979). Linear Algebra and Geometry. pp. 45. ISBN 0521293243.
In mathematics, an emptyproduct, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative...
combination of B. The empty sum convention allows the zero-dimensional vector space V={0} to have a basis, namely the empty set. Emptyproduct Iterated binary...
The empty set is a subset of A The union of A with the empty set is A The intersection of A with the empty set is the empty set The Cartesian product of...
The constant 1 {\displaystyle 1} is a monomial, being equal to the emptyproduct and to x 0 {\displaystyle x^{0}} for any variable x {\displaystyle x}...
2\times 1=120.} The value of 0! is 1, according to the convention for an emptyproduct. Factorials have been discovered in several ancient cultures, notably...
(the empty sum) is 0, and the product of 0 numbers (the emptyproduct) is 1. The factorial 0! evaluates to 1, as a special case of the emptyproduct. The...
3=1+1+1} etc.). The product of 0 numbers (the emptyproduct) is 1 and the factorial 0! evaluates to 1, as a special case of the emptyproduct. Any number n...
factors of the product type. A parameterless constructor corresponds to the emptyproduct. If a datatype is recursive, the entire sum of products is wrapped...
_{k=0}^{n-1}(x+k).\end{aligned}}} The value of each is taken to be 1 (an emptyproduct) when n = 0 . These symbols are collectively called factorial powers...
9‼ = 9 × 7 × 5 × 3 × 1 = 945. The zero double factorial 0‼ = 1 as an emptyproduct. The sequence of double factorials for even n = 0, 2, 4, 6, 8,... starts...
where m = n, the value of the product is the same as that of the single factor xm; if m > n, the product is an emptyproduct whose value is 1—regardless...
the empty set, then ⟨ S ⟩ {\displaystyle \langle S\rangle } is the trivial group { e } {\displaystyle \{e\}} , since we consider the emptyproduct to be...
topology is the natural topology on the Cartesian product. Throughout, I {\displaystyle I} will be some non-empty index set and for every index i ∈ I , {\displaystyle...
that contains V , {\displaystyle V,} which has a product, called exterior product or wedge product and denoted with ∧ {\displaystyle \wedge } , such...
positive integers, including 1, by the convention that the emptyproduct is equal to 1 (the emptyproduct corresponds to k = 0). This representation is called...
as they do for positive integers b: The interpretation of b0 as an emptyproduct assigns it the value 1. The combinatorial interpretation of b0 is the...
the ADT corresponds to a product type similar to a tuple or record. A constructor with no fields corresponds to the emptyproduct (unit type). If all constructors...
unique empty diagram 0 → C. Since the empty category is vacuously a discrete category, a terminal object can be thought of as an emptyproduct (a product is...
{\alpha (\alpha -1)\cdots (\alpha -n+1)}{n!}}.} (If n = 0, this product is an emptyproduct and has value 1.) It converges for | x | < 1 {\displaystyle |x|<1}...
{\displaystyle \forall x\,P(x)\equiv \neg \exists x\,\neg P(x)} Empty sum and emptyproductEmpty function Paradoxes of material implication, especially the...
y\in S} . In other words, S is closed under taking finite products, including the emptyproduct 1. Equivalently, a multiplicative set is a submonoid of...
possible products of n {\displaystyle n} orthogonal basis vectors with indices in increasing order, including 1 {\displaystyle 1} as the emptyproduct, forms...
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