The role of the parentheses in the definition is to ensure that any formula can only be obtained in one way by following the inductive definition ( in other words, there is a unique parse tree for each formula ).
12.
A "'recursive definition "'( or "'inductive definition "') in mathematical logic and computer science is used to define the elements in a set in terms of other elements in the set ( Aczel 1978 : 740ff ).
13.
The inductive dimension of a topological space may refer to the " small inductive dimension " or the " large inductive dimension ", and is based on the analogy that )-dimensional } } balls have-dimensional boundaries, permitting an inductive definition based on the dimension of the boundaries of open sets.
14.
There is also an inductive definition of the natural product ( by mutual induction ), but it is somewhat tedious to write down and we shall not do so ( see the article on surreal numbers for the definition in that context, which, however, uses surreal subtraction, something which obviously cannot be defined on ordinals ).
15.
This is very similar to an inductive definition of lists of strings; the difference is that this definition specifies how to access the contents of the data structure namely, via the accessor functions head and tail and what those contents may be, whereas the inductive definition specifies how to create the structure and what it may be created from.
16.
This is very similar to an inductive definition of lists of strings; the difference is that this definition specifies how to access the contents of the data structure namely, via the accessor functions head and tail and what those contents may be, whereas the inductive definition specifies how to create the structure and what it may be created from.
17.
Inductive definitions require their " definientia " to be " positive ", in the sense that " definienda " can only appear in " definientia " under an even number of negations . ( This assumes that negation, conjunction, disjunction, and the universal quantifier are the primitive logical connectives, and the remaining classical connectives are simply defined symbols . ) The definition
18.
Proof by induction . . . corresponds immediately to this mode of generating the numbers . "'Definition by induction "'( not to be confused with'inductive definition'. . . ) is the analogous method of defining a number-theoretic function ? ( y ) or predicate P ( y ) . [ A number-theoretic function or predicate takes as its variables only a selection from the natural numbers and produces only a single natural number in turn ].
