Description
1. Rudin, Ch 1, # 1, 4, and 5.
2. Let F be a field and x, y and elements of F. Prove the following using only the field axioms
and the property of cancellation.
(i) If x + y = x then y = 0 (The additive identity is unique)
(ii) If x + y = 0 then y = −x (The additive inverse is unique)
(iii) If x 6= 0 and xy = x then y = 1 (The multiplicative identity is unique)
(iv) If x 6= 0 and xy = 1 then y = 1/x (The multiplicative inverse is unique)
3. Let F be an ordered field and x, y, z ∈ F arbitrary. Prove the following cancellation laws
(a) If x + y < x + z then y < z.
(b) If xy < xz and x > 0, then y < z .
1 Math 3137
