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# Empty product

In mathematics, an empty product, or nullary product, is the result of multiplying no factors. [1]

55 relations: Addition, Arithmetic, Arity, Binomial series, Binomial theorem, Binomial type, Cardinality, Cartesian product, Categorification, Category (mathematics), Category of groups, Category of rings, Category of sets, Classical logic, Coproduct, Diagram (category theory), Discrete category, Dual (category theory), Edsger W. Dijkstra, Empty set, Empty sum, Factorial, Falling and rising factorials, Finite difference, First-order logic, Function (mathematics), Fundamental theorem of arithmetic, Identity element, Identity function, Identity matrix, Infix notation, Initial and terminal objects, Iterated binary operation, Jaroslav Nešetřil, Jiří Matoušek (mathematician), König's theorem (set theory), Limit (category theory), Linear map, Lisp (programming language), Logical conjunction, Mathematical induction, Mathematics, Multiplication, Product (category theory), Python (programming language), S-expression, Singleton (mathematics), Stirling number, Tuple, Universal quantification, ... Expand index (5 more) »

Addition (often signified by the plus symbol "+") is one of the four basic operations of arithmetic; the others are subtraction, multiplication and division.

## Arithmetic

Arithmetic (from the Greek ἀριθμός arithmos, "number") is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them—addition, subtraction, multiplication and division.

## Arity

In logic, mathematics, and computer science, the arity of a function or operation is the number of arguments or operands that the function takes.

## Binomial series

In mathematics, the binomial series is the Maclaurin series for the function f given by f(x).

## Binomial theorem

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.

## Binomial type

In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities Many such sequences exist.

## Cardinality

In mathematics, the cardinality of a set is a measure of the "number of elements of the set".

## Cartesian product

In set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets.

## Categorification

In mathematics, categorification is the process of replacing set-theoretic theorems by category-theoretic analogues.

## Category (mathematics)

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is an algebraic structure similar to a group but without requiring inverse or closure properties.

## Category of groups

In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms.

## Category of rings

In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (preserving the identity).

## Category of sets

In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets.

## Classical logic

Classical logic (or standard logic) is an intensively studied and widely used class of formal logics.

## Coproduct

In category theory, the coproduct, or categorical sum, is a category-theoretic construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces.

## Diagram (category theory)

In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory.

## Discrete category

In mathematics, in the field of category theory, a discrete category is a category whose only morphisms are the identity morphisms: Since by axioms, there is always the identity morphism between the same object, we can express the above as condition on the cardinality of the hom-set Some authors prefer a weaker notion, where a discrete category merely needs to be equivalent to such a category.

## Dual (category theory)

In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite category Cop.

## Edsger W. Dijkstra

Edsger Wybe Dijkstra (11 May 1930 – 6 August 2002) was a Dutch systems scientist, programmer, software engineer, science essayist, and early pioneer in computing science.

## Empty set

In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

## Empty sum

In mathematics, an empty sum, or nullary sum, is a summation where the number of terms is zero.

## Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, The value of 0! is 1, according to the convention for an empty product.

## Falling and rising factorials

In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, (A reprint of the 1950 edition by Chelsea Publishing Co.) rising sequential product, or upper factorial) is defined as The value of each is taken to be 1 (an empty product) when n.

## Finite difference

A finite difference is a mathematical expression of the form.

## First-order logic

First-order logic—also known as first-order predicate calculus and predicate logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## Fundamental theorem of arithmetic

In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.

## Identity element

In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with them.

## Identity function

Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument.

## Identity matrix

In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere.

## Infix notation

Infix notation is the notation commonly used in arithmetical and logical formulae and statements.

## Initial and terminal objects

In category theory, a branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object (also called terminal element): T is terminal if for every object X in C there exists a single morphism X → T. Initial objects are also called coterminal or universal, and terminal objects are also called final.

## Iterated binary operation

In mathematics, an iterated binary operation is an extension of a binary operation on a set S to a function on finite sequences of elements of S through repeated application.

## Jaroslav Nešetřil

Jaroslav (Jarik) Nešetřil (born March 13, 1946 in Brno) is a Czech mathematician, working at Charles University in Prague.

## Jiří Matoušek (mathematician)

Jiří (Jirka) Matoušek (10 March 1963 – 9 March 2015) was a Czech mathematician working in computational geometry and algebraic topology.

## König's theorem (set theory)

In set theory, König's theorem states that if the axiom of choice holds, I is a set, \kappa_i and \lambda_i are cardinal numbers for every i in I, and \kappa_i for every i in I, then The sum here is the cardinality of the disjoint union of the sets mi, and the product is the cardinality of the Cartesian product.

## Limit (category theory)

In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits.

## Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

## Lisp (programming language)

Lisp (historically, LISP) is a family of computer programming languages with a long history and a distinctive, fully parenthesized prefix notation.

## Logical conjunction

In logic, mathematics and linguistics, And (∧) is the truth-functional operator of logical conjunction; the and of a set of operands is true if and only if all of its operands are true.

## Mathematical induction

Mathematical induction is a mathematical proof technique.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Multiplication

Multiplication (often denoted by the cross symbol "×", by a point "⋅", by juxtaposition, or, on computers, by an asterisk "∗") is one of the four elementary mathematical operations of arithmetic; with the others being addition, subtraction and division.

## Product (category theory)

In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces.

## Python (programming language)

Python is an interpreted high-level programming language for general-purpose programming.

## S-expression

In computing, s-expressions, sexprs or sexps (for "symbolic expression") are a notation for nested list (tree-structured) data, invented for and popularized by the programming language Lisp, which uses them for source code as well as data.

## Singleton (mathematics)

In mathematics, a singleton, also known as a unit set, is a set with exactly one element.

## Stirling number

In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems.

## Tuple

In mathematics, a tuple is a finite ordered list (sequence) of elements.

## Universal quantification

In predicate logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all".

## Vacuous truth

In mathematics and logic, a vacuous truth is a statement that asserts that all members of the empty set have a certain property.

In mathematics and in computer programming, a variadic function is a function of indefinite arity, i.e., one which accepts a variable number of arguments.

## Zero to the power of zero

Zero to the power of zero, denoted by 00, is a mathematical expression with no obvious value.

## 0

0 (zero) is both a number and the numerical digit used to represent that number in numerals.

## 1

1 (one, also called unit, unity, and (multiplicative) identity) is a number, numeral, and glyph.

## References

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