Definition

1. Noun:
   • (Mathematics): A definition of a function or a concept that is expressed in terms of itself, allowing its values or instances to be calculated or generated in a finite number of steps. It typically consists of a base case (a simple, non-recursive starting point) and a recursive rule (which defines more complex cases in terms of simpler ones).

Usage Examples

• Noun:
  • The factorial function is often given as a classic example of a recursive definition.
  • To understand the sequence, you must first examine its recursive definition.
  • The textbook provides a recursive definition for the set of all even numbers.

Advanced Usage

• In Formal Logic & Computer Science: A recursive definition is fundamental for defining data structures (like linked lists or trees) and algorithms (like quicksort or depth-first search).
  • The recursive definition of a binary tree states that it is either empty or consists of a node with two child binary trees.
• Self-Referential Nature: The power of a recursive definition lies in its ability to define an infinite set or process using a finite statement.
  • The recursive definition of the Fibonacci sequence elegantly captures its infinite nature in just two lines.

Variants and Related Words

• Recursive (adj): Relating to or involving the repeated application of a rule, definition, or procedure to successive results.
  • A recursive function calls itself.
• Recursively (adv): In a recursive manner.
  • The set is defined recursively.
• Recursion (n): The process of defining something in terms of itself; the use of a recursive definition or procedure.
  • The concept of recursion is central to this programming paradigm.

Synonyms

• Inductive definition: Often used interchangeably in mathematics, emphasizing definition from a base case.
• Self-referential definition: Highlights the aspect of the definition referring to itself.

Related Phrases

• Base case: The part of a recursive definition that provides a direct, non-recursive answer for the simplest input.
  • Every valid recursive definition must have a well-specified base case to terminate the process.
• Recursive step/rule: The part of the definition that expresses the solution for a given input in terms of solutions for smaller or simpler inputs.
  • The recursive step reduces the problem towards the base case.

Noun

1. (mathematics) a definition of a function from which values of the function can be calculated in a finite number of steps