>>11686895I can't honestly think of anything other than showing how associativity and commutativity are independent properties. Clearly associativity does not imply commutativity, but it requires reducing to magmas to show that commutativity does not imply associativity. If we consider the rock-paper-scissors game, denote the options by R, P and S, and define the outcome of the operation X*Y to be the winning choice of X and Y. Then we have:
RR = RS = SR=R
PP = PR = RP=P
SS = SP = PS=S
from which we see that this RPS-magma is commutative, but
(RP)S = PS=S
R(PS)=RS=R,
and hence not associative.
>>11686913>Assuming you're in the USNope.
>>11686914I'll let others fill in the details should you have some other questions after this, but let's address your idea that you can multiply undefined by anything. Since this "undefined" is not a number, we would need to extend multiplication somehow to make this concept work, and this would have to be consistent with the multiplication of actual numbers. Basically, there would be two possibilities:
(1) multiplying undefined by distinct numbers would give distinct values;
(2) undefined would absorb every number like 0 does.
Assuming consistency, we could only have one value for the product of a number and undefined, and this leads to those two possibilities. In either case, 0*undef would give precisely one outcome, so we would not get all numbers.
Nighty night~
