Outline:

• How to Write a Proof
• Proofs on Numbers
• Universal and Existential Statements
• Variable Ownership

Proofs

• what is a proof: a dialog between a proof writer and proof reader
  • proof writer knows a mathematical fact
  • proof reader is honest but skeptical
  • proof writer’s job is to take the reader on a journey from ignorance to understanding
• mathematical proofs consist of three things:
  • definitions: what terms are used in this proof? what do they formally mean?
  • intuitions: what does this theorem mean? why, intuitively, should it be true?
  • conventions: what is the standard format for writing a proof? what are the techniques for doing so?

First Proof

• Our theorem: if $n$ is an even integer, then $n^2$ is even.
• Definitions:
  • Integers → defined last class
  • Even → an integer $n$ is called even if there is an integer $k$ where $n=2k$. This definition means $0$ is even, negative numbers can be even, etc.

Digression: how to use a definition, example:

• if $n$ is even → we know that there is some $k \in \mathbb Z$ where $n=2k$
• to prove n is even → we must show that there is some $k \in \mathbb Z$ where $n=2k$

continue the proof:

• Intuitions:
  • Examples: $4$ is even, $4^2=16$ is even. $-2$ is even, $(-2)^2 = 4$ is even.
  • Pictures: using $n^2$ unit squares arranged in a square, we know one side can be split into $n=2k$, so we can also split the square down in the middle, like in $n^2 = 2 (2k^2)$
• Writing the actual proof with conventions:
  • Assume $n$ is an even integer. We want to show that $n^2$ is even.
  • Since $n$ is even, there is some integer $k$ such that $n=2k$. This means that $n^2 = (2k)^2 = 4k^2 = 2(2k^2)$
  • From this, we see that there is an integer $m$ (namely, $2k^2$) where $n^2=2m$. Therefore, $n^2$ is even, which is what we wanted to show. $\blacksquare$