The commutative property implies that x*y=y*x
x*y=1+x+y (by definition), and similarly y*x=1+y+x.
Therefore, 1+x+y=1+y+x. In either of these expressions the commutative property applies so:
1+x+y=1+y+x=x+1+y=y+1+x=y+x+1=x+y+1.
To prove associativity, let’s see if (w*x)*y=w*(x*y).
w*x=1+w+x, (w*x)*y=1+w*x+y=1+1+w+x+y=2+w+x+y.
x*y=1+x+y, so w*(x*y)=1+w+1+x+y=2+w+x+y.
Therefore, (w*x)*y=w*(x*y), and x*y is associative.
