دانلود کتاب رویکردی یکپارچه از جبر انتزاعی

Chapter 1: A Potpourri of Preliminary Topics

1.1. What Are Definitions, Axioms, and Proofs?

The Role of Definitions, Axioms, and Proofs in Higher Mathematics: Since at least the time of Euclid, circa 300 BC, the ultimate test of mathematical rigor has resided in the construction of proofs of mathematical statements. Without getting into deep matters of philosophy, a proof is a sequence of steps that starts with a known fact and ends with the desired final statement. Each step is required to follow logically from a combination of one or more of the following:
• Steps in the proof that have already been completed.
• Statements that have previously been proven.
• Axioms, which are statements that are assumed to be true.
• Definitions, which describe the properties possessed by objects.