I am self-studying the wonderful book, Elementary Geometry from an Advanced Standpoint.
In chapter 1, problem 19 it says: Suppose that the elements of R were
1, with addition and multiplication defined by these tables. Which of the postulates are true?
+ | 0 1 ---|------ 0 | 0 1 1 | 1 0 . | 0 1 ---|------ 0 | 0 0 1 | 0 1
One of the postulates is this:
A-4 For every
a in R there is exactly one number
-a in R, called the negative of
a, such that
a + (-a) = (-a) + a = 0
Obviously there are no negative numbers in the above tables. Thus I might argue that A-4 does not hold.
On the other hand, from some responses to other questions that I have read, in mathematics it is not the particular symbols that matter, but rather that the symbols (whatever they may be) exhibit the desired properties such as commutativity and associativity. Do I understand correctly? This is a real epiphany for me.
Thus, for each number in the above tables there is indeed a number that when added to the number yields
0 + 0 = 0 1 + 1 = 0
Thus I will now argue that A-4 does hold.
Which is correct, A-4 holds or A-4 does not hold?