Induction and Recursion Concepts Review

This set of flashcards covers key concepts related to induction, recursion, and fundamental properties used in mathematical proofs and definitions in discrete mathematics.

What are the two main parts of a proof by mathematical induction?

1. Basis Step: Show that the statement holds for the initial value (usually 1). 2. Inductive Step: Show that if the statement holds for an integer k, it also holds for k + 1.

What key property allows us to conclude that a statement is true for all positive integers in mathematical induction?

The well-ordering property, which states that every nonempty subset of the positive integers has a least element.

What is strong induction?

A form of mathematical induction where you assume the statement is true for all integers up to k to prove it for k + 1.

What does it mean for a set to be defined recursively?

A set is defined recursively when it includes a basis step (initial elements) and a recursive step (rules to generate new elements from existing ones).

How do you prove a result about recursively defined sets?

By using structural induction, which includes showing the basis step for initial elements and that if the property holds for certain elements, it holds for newly generated elements.

What is a well-formed formula in propositional logic?

A syntactically correct expression built using propositional variables and logical operators that follows specific formation rules.

How do you define the height of a binary tree recursively?

1. Basis Step: The height of a tree with only a root is 0. 2. Recursive Step: The height is 1 plus the maximum height of its left and right subtrees.

Define the Fibonacci numbers in terms of recursion.

f0 = 0, f1 = 1, fn = fn-1 + fn-2 for n >= 2.

What is the Euclidean algorithm used for?

It is used to find the greatest common divisor (gcd) of two integers.

What is structural induction?

A method for proving properties of recursively defined structures, focusing on the basis and recursive steps.

Explain how the principle of induction can be generalized to other sets.

By using a similar approach to induction on nonnegative integers, the well-ordering property can be used on ordered pairs or higher-dimensional sets.