applying the lim operator independently to both operands of the fraction, is a so-called "indeterminate form". That does not mean that the limit sought is necessarily undefined; rather, it means that the limit of , if it exists, must be found by another method, such as l'Hôpital's rule. The sum of 0 numbers (the empty sum) is 0, and the product of 0 numbers (the empty product) is 1. The factorial 0! evaluates to 1, as a special case of the empty product. Other branches of mathematics In set theory, 0 is the cardinality of the empty set: if one does not have any apples, then one has 0 apples. In fact, in certain axiomatic developments of mathematics from set theory, 0 is defined to be the empty set. When this is done, the empty set is the von Neumann cardinal assignment for a set with no elements, which is the empty set. The cardinality function, applied to the empty set, returns the empty set as a value, thereby assigning it 0 elements. Also in set theory, 0 is the lowest ordinal number, corresponding to the empty set viewed as a well-ordered set. In propositional logic, 0 may be used to denote the truth value false. In abstract algebra, 0 is commonly used to denote a zero element, which is a neutral element for addition (if defined on the structure under consideration) and an absorbing element for multiplication (if defined). In lattice theory, 0 may denote the bottom element of a bounded lattice. In category theory, 0 is sometimes used to denote an initial object of a category. In recursion theory, 0 can be used to denote the Turing degree of the partial computable functions. Related mathematical terms A zero of a function f is a point x in the domain of the function such that . When there are finitely many zeros these are called the roots of the function. This is related to zeros of a holomorphic function. The zero function (or zero map) on a domain D is the constant function with 0 as its only possible output value, i.e., the function f defined by for all x in D. The zero function is the only function that is both even and odd. A particular zero function is a zero morphism in category theory; e.g., a zero map is the identity in the additive group of functions.