Math 3137 – Homework 1 Solution

Description

1. Rudin, Ch 1, # 1, 4, and 5.

2. Let F be a eld and x; y and elements of F. Prove the following using only the eld axioms and the property of cancellation.

1. 1. 1. If x + y = x then y = 0 (The additive identity is unique)

1. 1. 2. If x + y = 0 then y = x (The additive inverse is unique)

1. 1. 3. If x 6= 0 and xy = x then y = 1 (The multiplicative identity is unique)

1. 1. 4. If x 6= 0 and xy = 1 then y = 1=x (The multiplicative inverse is unique)

3. Let F be an ordered eld and x; y; z 2 F arbitrary. Prove the following cancellation laws

1. 1. If x + y < x + z then y < z.

1. 2. If xy < xz and x > 0, then y < z .

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