I had a problem recently as part of class work that dealt with sets and the conditions imposed by them. This is part of a larger question that I simplified things down to. The parts I'm concerned with are
- Is $+$ associative?
- Is $\times$ distributive with respect to $+$?
I've tried several combinations and can't find anything that shows it's not true. However, some of my classmates argue that there is in fact a combination that shows it is not true.
Here are the addition and multiplication tables I ended up with. Is there any combination that can act as a counterexample? (I guess off the bat, you can eliminate combinations involving $0$s):
$$ \begin{array}{c|cccc} + & 0 & 1 & 2 & 3 \\ \hline 0 & 0 & 1 & 2 & 3 \\ 1 & 1 & 2 & 3 & 2 \\ 2 & 2 & 3 & 2 & 1 \\ 3 & 3 & 2 & 1 & 0 \end{array} \qquad \begin{array}{c|cccc} \times & 0 & 1 & 2 & 3 \\ \hline 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 \\ 2 & 0 & 2 & 2 & 0 \\ 3 & 0 & 3 & 0 & 3 \end{array} $$
$$1+(1+3)=1+2=3,\quad\text{but}\quad(1+1)+3=2+3=1$$
$$1+(2+3)=1+1=2,\quad\text{but}\quad(1+2)+3=3+3=0$$
There are probably others.
$$2(1+2)=2(3)=0,\quad\text{but}\quad2(1)+2(2)=2+2=2$$
Again, there are probably others.