Recursion vs. Induction in Discrete Mathematics

• "Recursion" is a way of defining some mathematical object (including a function or computation whose definition involves a recursive algorithm);
• "Induction" is a way of proving some mathematical statement.

Extremely often, if a mathematical statement is made about a recursively-defined object, then the proof of that statement will involve induction.

https://math.stackexchange.com/questions/3779213/what-is-the-difference-between-recursion-and-induction

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In discrete mathematics, recursion and induction are both fundamental concepts, but they serve different purposes and are used in different contexts:

1. Recursion:
   • Definition: Recursion is a method of defining sequences, functions, or other mathematical constructs by specifying a starting point (base case) and a rule for how each successive term is derived from the previous ones (recursive step). It's a way of breaking down a problem into smaller, more manageable parts.
   • Use in Mathematics: Recursion is often used in algorithms, where a function calls itself with modified parameters. It is a powerful tool in computer science for solving problems that can be broken down into simpler, similar subproblems.
   • Example: A classic example is the Fibonacci sequence, where each term is the sum of the two preceding ones. The recursive definition is: F(0)=0, F(1)=1, and F(n)=F(n−1)+F(n−2) for n>1.
2. Induction:
   • Definition: Induction is a method of mathematical proof typically used to prove statements about natural numbers. It consists of two steps: the base case, where the statement is verified for the initial value (usually 0 or 1), and the inductive step, where one assumes the statement is true for some arbitrary natural number kk and then proves it for k+1.
   • Use in Mathematics: Induction is a fundamental proof technique in mathematics. It is especially useful for proving properties or formulas that hold for all natural numbers.
   • Example: Proving that the sum of the first nn natural numbers is n(n+1)/2 can be done using induction. The base case is verified forn=1, and then assuming the formula holds for somen=k, it is proven for n=k+1.

In summary, recursion is a method for defining and solving problems by breaking them into smaller instances of the same problem, while induction is a technique for proving that a certain property or statement is true for all cases in a defined set, typically the natural numbers.