Mathematical Induction: Standard, Strong, and Structural Methods

All flashcards

Mathematical induction

A deductive method that proves a starting case and shows each case implies the next.

Induction hypothesis

The temporary assumption that $P(k)$ holds, used to prove $P(k+1)$.

Strong induction

A method that permits assuming the proposition holds for every earlier relevant case.

Structural induction

A method that proves properties of recursively defined objects by following their construction rules.

Transfinite induction

An induction method over well-founded orders, assuming the property for all smaller elements.

Quiz questions

1. In an induction proof of $0+1+\cdots+n=\frac{n(n+1)}{2}$, what expression results after adding $k+1$ to the sum through $k$ and simplifying with the hypothesis?Answer: $\frac{(k+1)(k+2)}{2}$
2. To prove $|\sin((k+1)x)|\leq(k+1)|\sin x|$ from $|\sin(kx)|\leq k|\sin x|$, which tools are used?Answer: The angle-addition identity, the triangle inequality, and $|\cos t|\leq 1$.
3. An argument claims all horses have the same color by comparing two overlapping groups of $n$ horses. Why does its induction step fail when $n=1$?Answer: The two groups of one horse are separate, so they have no horse in common.
4. In strong induction, what may be assumed when proving the case $P(k)$?Answer: The proposition holds for every earlier relevant value.
5. How does structural induction prove a property of recursively defined trees?Answer: Verify it for basic trees and show each construction rule preserves it.