## Section0.2Definitions, Theorems, and Conjectures

If we think of mathematics as a building, then definitions provide the foundation, theorems are the bricks, and logic is the mortar that connects them and holds them together.

Definitions introduce terminology to define mathematical objects and properties.

Theorems are statements about defined objects. A theorem uses defined terms and is derived from a sequence of logical arguments using definitions and other, previously proven theorems. To prove a theorem is to construct a sequence of logical arguments that make it a true statement (there can be more than one such sequence). The sequence of logical arguments used to derive the theorem is called a proof of the theorem.

In this course we do not expect you to come up with new theorems or to be able to prove known theorems. Nevertheless we will prove most theorems in these notes, if only to show you that everything follows from the definitions in a sequence of logical steps. Proofs of theorems are either given after the theorems (they start with Proof.) or the argument for the correctness is given before the statement.

Although it is possible to give definitions of the integers and their arithmetic and to prove their properties, we will assume familiarity with them.

We will also encounter statements that are believed to be true, but nobody has been able to find a proof yet. These statements are called conjectures.