Suppose we have a field F = {0, 1}

1 + 1 = ?

How does one determine if 1+1 = 1 or if 1+1 = 0?

Can I define it to be one of the two choices?

Printable View

- Jan 15th 2011, 06:20 PMNoxideAddition on a field with two elements...
Suppose we have a field F = {0, 1}

1 + 1 = ?

How does one determine if 1+1 = 1 or if 1+1 = 0?

Can I define it to be one of the two choices? - Jan 15th 2011, 06:27 PMtonio
- Jan 15th 2011, 06:35 PMNoxide
Ah yes, silly of me to ask. Thanks tonio.

1 + 1 = 1

1 + [1 + (-1)] = 1 + (-1)

1 + 0 = 0

1 = 0 :( - Jan 16th 2011, 07:04 AMHallsofIvy
Another way of saying the same thing is that every element of a field must have an

**additive inverse**. The additive inverse of 1 cannot be 0 because $\displaystyle 1+ 0= 1\ne 0$. Since 0 and 1 are the only members of the field, 1 must be its own inverse: 1+ 1= 0.