Mathematical Writing

• Follow guidelines for use of symbols.
• Use displays for important formulas and long expressions and calculations.
• Use effective transition phrases to connect the stages of a proof.

Proof Techniques

• Use counterexamples to disprove universally quantified statements.
• Construct proofs by contradiction.
• Construct proofs by induction.

Sets and Set Operations

• Construct proofs about set inclusion and set equality using
  • the algebra of sets
  • "element-chasing".
• Define the Cartesian product of sets and use Cartesian products in proofs.

The Natural Numbers N and the Integers Z

• State the definition of divisibility and use it in proofs.
• State the definition of congruence and use it in proofs.
• State the well-ordering property.
• State the definition of binomial coefficients and use
  • the "Pascal triangle" rule;
  • the subset counting property;
  • the factorial formula.
• Use induction to define sequences recursively.

Fields: the Rational Numbers Q and the Real Numbers R

• State the definitions of Field and Ordered Field.
• Use elementary properties of ordered fields to prove statements about rational and real numbers.
• State the Triangle Inequality, and use it in proofs.

alex at iastate dot edu

Document last modified on Wed Mar 18 2009