[Solved] MATH3137-Homework 1

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.
   • If x + y = x then y = 0 (The additive identity is unique)
   • If x + y = 0 then y = x (The additive inverse is unique)
   • If x 6= 0 and xy = x then y = 1 (The multiplicative identity is unique)
   • 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 Prove the following cancellation laws
   • If x + y < x + z then y < z.

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