First of all, the definition of loop is different from that of recursion. Loop definition is defined by the relationship between the elements of the set and the set itself, while recursion is defined by the call of the function or expression itself. Specifically, the definition of a cycle usually takes the following form: for any X belonging to a set A, if there is a Y belonging to A, so as to meet certain conditions, it is said that X has certain properties. Recursion can be expressed as follows: For any N belonging to natural number set N, if an M belongs to N, so that certain conditions are met, then N is said to have certain properties.
Secondly, the definition of cycle and the nature of recursion are also different. The definition of cycle usually has reflexivity, symmetry and transitivity, while recursion does not necessarily have these properties. For example, for a set A={ 1, 2,3}, we can define its cyclic subsets as {1}, {2}, {3}, {1, 2}, {1, 3}, {2} This definition is reflexive, symmetrical and transitive, so it is a circular definition. However, if we recursively define a subset of this set, we don't necessarily have these attributes.
Finally, the usage scenarios of loop definition and recursion in practical applications are also different. The definition of cycle is usually used to describe the relationship between sets or the characteristics of a certain structure, such as the cycle and tree structure in graph theory. Recursion is more used in algorithm design and computer science, such as Fibonacci sequence, quick sorting and so on.
To sum up, although loop definition and recursion are commonly used concepts in mathematics, there are some obvious similarities and differences between them. Understanding these similarities and differences will help us better understand and apply these two concepts.