Prove that ∗ defined on ℤ by x ∗ y = 1 + x + y is commutative and associative.

Question

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Answer 1

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.