A1: a + b = b + a
A2: left( a + b right) + c = a + left( b + c right)
A3: a + 0 = a
A4: a + left( -a right) = 0
M1: ab = ba
M2: a left( bc right) = left( ab right) c
M3: a cdot 1 = a
M4: a cdot a^{-1} = 1
D1: a left( b+ c right)= ab + ac
Theorems Proved on the next page:
- 0 cdot a = 0
- If a + b = a then b = 0
- – left( -a right) = a
- If a + b = 0 then a = -b
- – left( a + b right) = left( -a right) + left( -b right)
- left( -a right) left( -b right) = ab