### 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;
• 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.