Empty product

In mathematics, an empty product, or nullary product, is the result of multiplying no numbers. Its numerical value is 1, the multiplicative identity, just as the empty sum—the result of adding no numbers—is zero, or the additive identity.

The empty product is used in discrete mathematics, algebra, the study of power series, and computer programs.

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; these are discussed below.

Nullary arithmetic product

Frequent examples

Two often-seen instances are a⁰ = 1 (any number raised to the zeroth power is one) and 0! = 1 (the factorial of zero is one).

More examples of the use of the empty product in mathematics may be found at the following pages: binomial theorem, factorial, fundamental theorem of arithmetic, birthday paradox, Stirling number, König's theorem, binomial type, difference operator, Pochhammer symbol, proof that e is irrational, prime factor, binomial series, multiset.

Intuitive justification

Imagine a calculator that can only multiply. It has an "ENTER" key and a "CLEAR" key. One would wish that, for example, if one presses "CLEAR", 7 "ENTER", 3 "ENTER", 4 "ENTER", then the display reads 84, because 7 × 3 × 4 = 84. More precisely, we specify:

• A number is displayed just after pressing "CLEAR";
• When a number is displayed and one enters another number, the product is displayed;
• Pressing "CLEAR" and entering a number results in the display of that number.

Then the starting value after pressing "CLEAR" has to be 1. After one has pressed "clear" and done nothing else, the number of factors one has entered is zero. Therefore the product of zero numbers is 1.

Logarithms

The definition of an empty product can be based on that of the empty sum:

The sum of two logarithms is equal to the logarithm of the product of their operands, i.e. for any base b>0:

$\log _{b}n+\log _{b}m=\log _{b}nm$

and

$b^{\log _{b}n+\log _{b}m}=nm$

and more generally

$\prod _{i}x_{i}=b^{\sum _{i}\log _{b}x_{i}}$

i.e., multiplication across all elements of a set is b to the power of the sum of all logarithms of the set's elements.

Using this property as definition, and extending this to the empty product, the right-hand side of this equation evaluates to $b^{0}$ for the empty set, because the empty sum is defined to be zero, and therefore the empty product must equal one.

0 raised to the 0th power

For more information, see the section on Zero to the zero power in the article on exponentiation.

In set theory and combinatorics, the cardinal number n^m is the size of the set of functions from a set of size m into a set of size n. If m is positive and n is zero, then there are no such functions, because there are no elements in the latter set to map those of the former set into. Thus 0^m = 0 when m is positive. However, if both sets are empty (have size 0), then there is exactly one such function — the empty function. For this reason, authors in combinatorics and set theory frequently define 0⁰ to be 1 when it represents an empty product.

Nullary intersection

For similar reasons, the intersection of an empty set of subsets of a set X is conventionally equal to X. See nullary intersection for more information.

Nullary Cartesian product

Consider the general definition of the Cartesian product:

$\prod _{i\in I}X_{i}=\{f:I\to \bigcup _{i\in I}X_{i}\ |\ \forall i\ f(i)\in X_{i}\}.$

If I is empty, the only satisfying f is the empty function:

$\prod \varnothing =\{f_{\varnothing }:\varnothing \to \varnothing \}.$

Thus, the cardinality of the Cartesian product of no sets is 1.

Under the perhaps more familiar n-tuple interpretation,

$\prod \varnothing =\{()\},$

that is, the singleton set containing the empty tuple. Note that in both representations the empty product has cardinality 1.

Nullary Cartesian product of functions

The empty Cartesian product of functions is again the empty function.

Nullary categorical product

In any category, the product of an empty family is a terminal object of that category. This can be demonstrated by using the limit definition of the product. An n-fold categorical product can be defined as the limit with respect to a diagram given by the discrete category with n objects. An empty product is then given by the limit with respect to the empty category, which is ther terminal object of the category if it exists. This definition specialises to give results as above. For example, in the category of sets the categorical product is the usual cartesian product, and the terminal object is a singleton set in the category of groups the categorical product is the cartesian product of groups, and the terminal object is a trivial group with one element. To obtain the usual arithmetic definition of the empty product we must take the decategorification of the empty product in the category of finite sets.

Dually, the coproduct of an empty family is an initial object. Nullary categorical products or coproducts may not exist in a given category; e.g. in the category of fields, neither exists.

In computer programming

Most programming languages do not permit the direct expression of the empty product, because multiplication is taken to be an infix operator and therefore a binary operator. (A programmer may, of course, implement it.) Languages implementing variadic functions are the exception. For example, the fully parenthesized prefix notation of Lisp languages gives rise to a natural notation for nullary functions:

(* 2 2 2)   ; evaluates to 8
(* 2 2)     ; evaluates to 4
(* 2)       ; evaluates to 2
(*)         ; evaluates to 1

Many programming languages with infix multiplication also offer a generalized multiplication function, often called product, which can be applied to a list of numbers. Such functions return 1 when applied to an empty list.

See also

• Iterated binary operation

External links

• PlanetMath article on the empty product

Empty product at PlanetMath.