Commutative monoid proof

Question

Consider the algebraic structure $(Z × Z, ⊥)$, with $(a, b) ⊥ (c, d) = (a + c, bd)$ for each $a$, $b$, $c$, $d ∈ Z$.

Prove that $(Z × Z, ⊥)$ is a commutative monoid

My attempt:

Addition $(a+c)$ and multiplication $(bd)$ are commutative operations in the set of integer numbers, so $(Z × Z, ⊥)$ is a commutative monoid.

Is my proof right?

Answer 1

You proved correctly that $\perp$ is commutative. You must also provo that it is associative and that there is an identity in $(\mathbb{Z}\times\mathbb{Z},\perp)$.