🔢Lower Division Math Foundations Unit 6 – Mathematical Induction & Recursion

Key Concepts and Definitions

• Mathematical induction proves statements that depend on a natural number $n$
• Recursion defines a function or sequence in terms of itself, using previously calculated values
• Base case serves as the starting point for both induction and recursion
• Inductive step assumes the statement holds for $n = k$ and proves it for $n = k + 1$
• Recursive step defines how to calculate the next value using the previous one(s)
• Well-ordering principle states that every non-empty set of positive integers has a least element
  • Forms the basis for mathematical induction
• Strong induction proves a statement for all integers greater than or equal to a starting value
  • Assumes the statement holds for all values less than or equal to $k$ and proves it for $k + 1$

Principles of Mathematical Induction

• Induction proves statements of the form $\forall n \in \mathbb{N}, P(n)$, where $P(n)$ is a predicate
• Two key components of an inductive proof are the base case and the inductive step
• Base case proves the statement holds for the smallest value of $n$ (usually 0 or 1)
• Inductive hypothesis assumes the statement holds for an arbitrary value $k$
• Inductive step proves that if the statement holds for $k$, it must also hold for $k + 1$
  • Establishes the truth of the statement for all natural numbers greater than the base case
• Principle of mathematical induction is based on the well-ordering principle of the natural numbers
• Induction is a powerful tool for proving statements involving summations, inequalities, and divisibility

Steps in Inductive Proofs

• State the proposition to be proved, typically in the form $\forall n \in \mathbb{N}, P(n)$
• Prove the base case, usually $P(0)$ or $P(1)$, by directly verifying the statement
• State the inductive hypothesis, assuming $P(k)$ holds for an arbitrary natural number $k$
• Prove the inductive step by showing that if $P(k)$ is true, then $P(k + 1)$ must also be true
  • Often involves algebraic manipulation or other mathematical techniques
• Conclude that by the principle of mathematical induction, $P(n)$ holds for all natural numbers $n$
• Clearly communicate each step of the proof and provide justifications when necessary
• Double-check the proof for logical consistency and correctness

Common Induction Examples

• Proving the sum of the first $n$ positive integers: $\sum_{i=1}^n i = \frac{n(n+1)}{2}$
• Proving the sum of consecutive odd numbers: $\sum_{i=1}^n (2i-1) = n^2$
• Proving divisibility properties, such as the divisibility of $n^3 - n$ by 6 for all $n \in \mathbb{N}$
• Proving inequalities, like the inequality $2^n < n!$ for all $n \geq 4$
• Proving the binomial theorem for natural number exponents using induction
• Proving the correctness of recursive algorithms, such as the Fibonacci sequence or factorial function
• Proving properties of geometric sums, like $\sum_{i=0}^n ar^i = a\frac{1-r^{n+1}}{1-r}$ for $r \neq 1$

Introduction to Recursion

• Recursion is a method of defining functions or sequences in terms of themselves
• A recursive definition consists of two parts: the base case(s) and the recursive step
• Base case provides a direct definition for the smallest input value(s)
  • Serves as a stopping condition to prevent infinite recursion
• Recursive step defines the function or sequence in terms of smaller input values
  • Expresses the current value in terms of previously calculated values
• Recursive algorithms break down a problem into smaller subproblems until the base case is reached
• Recursion can often lead to elegant and concise solutions to complex problems
• Understanding the flow of recursive calls and the stack is crucial for effective use of recursion

Recursive Algorithms and Functions

• Factorial function $n!$ is a classic example of a recursive function
  • Base case: $0! = 1$
  • Recursive step: $n! = n \times (n-1)!$ for $n > 0$
• Fibonacci sequence is another well-known example of recursion
  • Base cases: $F_0 = 0$ and $F_1 = 1$
  • Recursive step: $F_n = F_{n-1} + F_{n-2}$ for $n > 1$
• Recursive algorithms for searching and sorting, such as binary search and merge sort
• Recursive solutions to mathematical problems, like the Tower of Hanoi or calculating permutations
• Recursive traversal of data structures, such as trees and graphs
  • Depth-first search (DFS) and breadth-first search (BFS) are common recursive algorithms
• Recursive backtracking for solving optimization problems and puzzles (Sudoku, N-Queens)

Relationship Between Induction and Recursion

• Mathematical induction and recursion are closely related concepts
• Inductive proofs often involve recursive definitions or algorithms
• Recursive functions can be proved correct using mathematical induction
  • Base case corresponds to the base case of the inductive proof
  • Recursive step is verified using the inductive hypothesis
• Induction can be used to prove properties of recursively defined sequences or functions
  • Example: proving the closed-form formula for the Fibonacci sequence using induction
• Both induction and recursion rely on the well-ordering principle of the natural numbers
• Understanding the connection between induction and recursion enhances problem-solving skills

Applications and Problem-Solving Strategies

• Identify problems that exhibit a recursive structure or can be solved using induction
• Break down complex problems into smaller subproblems that can be solved recursively
• Determine appropriate base cases and recursive steps for a given problem
• Use mathematical induction to prove the correctness of recursive algorithms or formulas
• Analyze the time and space complexity of recursive algorithms using recurrence relations
• Apply memoization or dynamic programming techniques to optimize recursive solutions
  • Store previously calculated values to avoid redundant recursive calls
• Consider iterative alternatives to recursive solutions for improved efficiency
• Practice solving a variety of problems using induction and recursion to develop proficiency