ABSTRACT
This chapter is devoted to the study of deductive mathematical systems and elementary proof techniques. In Section 2.1, we define deductive mathematical systems and discuss the Euclidean and non-Euclidean geometries, the system of natural numbers, and the system of integers. In Section 2.2, basic techniques for proving conditional statements such as direct proof, proof by contraposition, proof by contradiction, and proof by cases are presented. We also discuss proving biconditional statements, proving a statement by contradiction, and proving statements which contain quantifiers. Next, we present some well-known mathematical conjectures and show how to prove and disprove conjectures. And finally, we define and examine the system of rational numbers and the system of real numbers.
