I am trying to prove associativity for the following.
Addition:
$$
\begin{array}{c|cccc}
+ & 0& 1& B & D \\
\hline
0& 0 & 1 & B & D \\
1 & 1 & 0 & D & B \\
B & B & D & 0 & 1 \\
D & D & B & 1 & 0 \end{array}
$$
I tried a proof by contradiction and it is technically correct but it is too long and tedious and I am wondering if there is a better way that I haven't thought of. I don't really want the answer just like a push in the right direction.
Also, just wondering if I should try to define addition in like an algebraic way but again I don't understand how that would work. Any help is appreciated.