# Lecture - Introduction to University Mathematics, Lecture 1

> Source: https://ollybritton.com/notes/uni/prelims/mt22/introduction-to-uni-maths/lectures/lecture-1/ · Updated: 2022-10-07 · Tags: uni, lecture

> Mainly about induction and proving simple things with induction.

### Notes
Principle of Mathematical Induction: Let $P(n)$ be a family of statements indexed by the natural numbers. Suppose $P(0)$ is true, and if $P(n)$ is true then $P(n+1)$ is true. Therefore $P(n)$ is true for all natural numbers $n$.

To use the inductive hypothesis in a prove is to use what you’re assuming to be true about $P(n)$.

Strong induction is where you rely on everything less than $n+1$ already being assumed true. Can prove by using normal induction on a new family of statements $Q(n)$ that is ‘$P(k)$ is true for $k = 0, 1, 2…n$ ’.

Example of a proof using strong induction, saying that every natural number greater than 1 can be expressed as a product of one or more primes.

Defining addition recursively by writing the addition of $m + (n+1)$ as $(m + n) + 1$ and then saying that $m + 0$ is just $m$.

Can use this recursive definition to prove associativity via induction.

Proving the well-ordering property of $\mathbb{N}$, “every non-empty subset of N contains a least element” using a mix of proof by contradiction and proof by induction. The gist is showing that the complement of the magical set with no least element must actually be all natural numbers.

### Flashcards
What is the Principle of Mathematical Induction?::
Let $P(n)$ be a family of statements indexed by the natural numbers. Suppose $P(0)$ is true, and if $P(n)$ is true then $P(n+1)$ is true. Then $P(n)$ is true for all natural numbers $n$.

What is the difference between normal induction and strong induction?::
In strong induction you rely on all previous statements rather than just the most immediate one

How can you prove strong induction?::
Use normal induction on $Q(n)$ being the statement ‘$P(k)$ is true for all $0, 1, 2…n$’

What is the inductive hypothesis?::
What you assume to be true about previous statements in an induction question.

What’s a simple example of a proof using strong induction?::
Proving that every natural number greater than 2 can be factorised into a product of primes.

What’s the two components of the recursive definition of addition?::
Saying there is an additive identity of $0$ and then $m + (n+1)$ is the same as $(m + n) + 1$

What’s the well-ordering property of $\mathbb{N}$?::
Every non-empty subset of $\mathbb{N}$ contains a least element.

What’s the gist of proving the well-ordering property of $\mathbb{N}$?::
Showing that complement set of the non-empty subset must contain all natural numbers by strong induction.

---
Olly Britton — https://ollybritton.com. Machine-readable index: https://ollybritton.com/llms.txt