Reading and Writing Proofs

As You Begin

Analyze the statement of the theorem:

• identify the hypothesis and conclusion;
• parse compound statements.

What proof method is/could be used?

• direct;
• contrapositive;
• contradiction;
• induction;
• etc.

Construct the outline of the proof. Key words in a formal proof indicate the structure of the outline:

• first line;
• last line;
• next to last line ("abstraction statement");
• subproofs, if any;
• then fill in the details.

Write the Paragraph Proof

• use complete, grammatical sentences;
• use appropriate notation, but never begin a sentence with a symbol;
• introduce each new variable properly;
• generally follow the proof outline;
• improve readability by omitting steps and reasons if your audience will tolerate it;
• do not ignore subtleties;
• do not explain how you discovered the proof;
• use "we," never "I";
• a new direction needs a new paragraph;
• preview a long or complicated argument;

Important Skills

• identify the logical forms of statements;
• recognize when two statements say the same thing and when they don't (truth table);
• remember definitions of technical terms;
• remember the meaning of notation;
• remember previous theorems;
• use quantifiers correctly;
• negate statements correctly;
• know how to define a set with a property;
• know how to represent sets of sets and use indices;
• recognize relations and operations and know the common ones;
• know what a function is and how to use function notation;
• remember there is more than one way to prove something.

fall 2007 course description
the mathematical perspective
homework guidelines
paper and talk guidelines
additional resources
truth table forms
back to logic & proof
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