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 .
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