# Lecture - Introduction to University Mathematics, I

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.