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Floyd's triangle is a triangular array of natural numbers used in computer science education. It is named after Robert Floyd . It is defined by filling the rows of the triangle with consecutive numbers, starting with a 1 in the top left corner:
For example, it is common to identify the truth value with the number and the truth value with the number . Once this identification has been made, the characteristic function of a set A {\displaystyle A} , which always returns 1 {\displaystyle 1} or 0 {\displaystyle 0} , can be viewed as a predicate that tells whether a number is in the set A ...
In computer science, a for-loop or for loop is a control flow statement for specifying iteration. Specifically, a for-loop functions by running a section of code repeatedly until a certain condition has been satisfied. For-loops have two parts: a header and a body. The header defines the iteration and the body is the code executed once per ...
The code above specifies a list of strings to be either empty, or a structure that contains a string and a list of strings. The self-reference in the definition permits the construction of lists of any (finite) number of strings. Another example of inductive definition is the natural numbers (or positive integers):
LOOP is a simple register language that precisely captures the primitive recursive functions. [1] The language is derived from the counter-machine model . Like the counter machines the LOOP language comprises a set of one or more unbounded registers , each of which can hold a single non-negative integer.
A total order on the natural numbers is defined by letting a ≤ b if and only if there exists another natural number c where a + c = b. This order is compatible with the arithmetical operations in the following sense: if a, b and c are natural numbers and a ≤ b, then a + c ≤ b + c and ac ≤ bc.
Loop-invariant code consists of statements or expressions that can be moved outside a loop body without affecting the program semantics. Such transformations, called loop-invariant code motion, are performed by some compilers to optimize programs. A loop-invariant code example (in the C programming language) is
In Set, the category of sets, the standard natural numbers are an NNO. [6] A terminal object in Set is a singleton, and a function out of a singleton picks out a single element of a set. The natural numbers 𝐍 are an NNO where z is a function from a singleton to 𝐍 whose image is zero, and s is the successor function.