I. Field Axioms - Exercises pp. 31-32 (Algebraic Structures)

This first section of the course focuses on the Real Number System and the Complex Number System. In our first several classes, we examine the general laws of behaviors or axioms for real numbers. The axioms are given in three sets; 1) the Field Axioms; 2) The Order Axioms; 3) The Completeness Axioms. The first section focuses on the field axioms (6 in number) from which we can deduce all familiar properties of real numbers that depend ONLY upon addition and multiplication.


Exercises

1. Use the technique of Theorem 12 to prove that (-a)(b = a(-b) = - (ab).
2. Prove that (-a)(-b) = ab. Hence deduce (-1)(-1) = 1.
3. Prove that -0 = 0.
4. Verify the following: i) -(a + b) = -a - b; ii) -(a-b) = -a + b; iii) -(-a+b) = a-b.