Mathematical Induction Lecture Notes

This set of flashcards covers key concepts related to mathematical induction as discussed in the lecture.

Mathematical Induction

A formal method of proof used to verify conjectures based on natural numbers, involving a base case and an inductive step.

Base Case

The initial step in mathematical induction, where the conjecture is verified for the smallest natural number, usually n=1.

Inductive Hypothesis

The assumption made in mathematical induction that the conjecture holds for an arbitrary case k.

Induction Step

The part of the proof in mathematical induction where it is shown that if the hypothesis holds for n=k, it must also hold for n=k+1.

Recursive Definition

A method of defining a function in terms of itself, typically involving a base case and one or more recursive cases.

Conjecture

A statement or proposition that is suspected to be true based on observations but has not yet been proven.

Predicate (p_k)

A statement that can be true or false depending on the values of its variables, often used in the context of mathematical induction.

Arithmetic Progression

A sequence of numbers in which the difference between consecutive terms is constant.

Proof by Deduction

A type of proof where conclusions are drawn from premises using logical reasoning.

Inductive Case

In mathematical induction, the scenario where the truth of the conjecture is established for n=k based on its truth for n=k-1.

Natural Numbers

The set of positive integers typically defined as {1, 2, 3, …}.

Formal Logic

A system of reasoning that uses formal structures and symbols.

Verification

The process of establishing the truth or accuracy of a statement or hypothesis.

Polynomial

A mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.