Writing Proofs

Return to List of Lectures

Previous Lecture

Misc

Flyer for research weekend if you want information you can take with you

Questions?

In a formal proof, can you write reasons for each step in a series of displayed equations beside the equations?

No, reasons/explanations have to be written in prose paragraphs.

Can know-show tables ever appear in a formal proof?

No, know-show charts are always scratch work.

Which doesn’t mean they aren’t important, just that their importance is for discovering proofs rather than for presenting them.

Writing Proofs

LaTeX

A tool for producing typeset-quality mathematics writing

Free LaTeX systems are available for all major computers and operating systems

Basic work flow

1. Write text document marked up with LaTeX
2. Process that text document with the LaTeX program
3. LaTeX gives you a nicely formatted PDF

Document concepts

• Documents consist of a preamble (sets up parameters for the rest of the document) and a body (the document text)
• The body may contain environments (sections subject to section-specific formatting, e.g., proofs, lists, etc.) and commands (instructions to LaTeX to do something), as well as regular text

Example: A very simple document that illustrates the basic structure and some of LaTeX’s capabilities.

The LaTeX source file

The PDF file that LaTeX produces

Example: the proof from last class that an even number plus an odd number is odd written up formally via LaTeX

Some relevant guidelines for formal proofs from recent readings and videos:

• State theorem, rewritten if necessary to not be posed as a question
• Proof starts right after theorem statement, labeled by the word “Proof” in some distinctive typeface (bold, italic, etc.)
• State assumptions and proof technique
• Prove the theorem (having notes from planning helps, so does keeping audience in mind)
  • Use complete sentences
  • Guide readers through the argument
  • Justify your steps
  • Use plural pronouns (e.g., us) not singular (e.g., I)
  • Italicize variables, boldface vectors
  • Display (center and set off from prose) important equations; align equality signs in series of displayed equations and don’t repeat identical left-hand sides
  • Keep mathematical symbols in equations, not in the prose
• State conclusion explicitly when the argument reaches it
• End with “QED” or similar end-of-proof symbol

The LaTeX source for this proof

The resulting PDF file

Problem Set 2

Practice writing formal proofs

See handout for details

Next

Introduction to propositional logic

Read section 2.1 of textbook

Next Lecture