Associative properties with two numbers

Question

I am working through a number theory text and I am given a set $S=\{A,B\}$ and it has the properties:

1) $A+A=A$
2) $B+B=A$
3) $A+B=B+A=B$
4) $A(A)=A$
5) $A(B)=B(A)= A$
6) $B(B)=B$

I am to verify this set conforms to the axioms of a field but the associative properties of addition and multiplication are defined using three numbers. How do I go about showing the set satisfies the associative properties given only two numbers? - thanks

Answer 1

The associative property isn't defined for three different elements, just three elements. For example:

$1+(1+1) = (1+1)+1$

So you need to verify, for example, that $A+(A+B)=(A+A)+B$, et cetera.