Addition on a field with two elements...

• Jan 15th 2011, 06:20 PM
Noxide
Addition 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 PM
tonio
Quote:

Originally Posted by Noxide
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?

There's no choice: it must be that \$\displaystyle 1+1=0\$ , otherwise we'd get \$\displaystyle 1=0\$ ...

Tonio
• Jan 15th 2011, 06:35 PM
Noxide
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 AM
HallsofIvy
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.