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### Problem 5

Determine whether the binary operation $*$ defined on $\mathbb{Q}$ by letting $a * b = ab/2$ is commutative. Is $*$ associative?

**Solution**

Since $a, b \in \mathbb{Q}$, we can write $a = \frac{p}{q}$ and $b = \frac{m}{n}$ where $p, q, m, n \in \mathbb{Z}$ and $q, n \neq 0$

Now, for an operation to be commutative $a * b = b * a$. $a = \frac{p}{q}$ and $b = \frac{m}{n}$ so we can substitute for $a$ and $b$, which yields,

(1)By properties of integer multiplication $a * b = b * a$. Hence, the operation is commutative.

To show the operation is associative, we must introduce another rational number, $c = \frac{x}{y}$ where $x, y \in \mathbb{Z}$ and $y \neq 0$.

For an operation to be associative $(a * b) * c = a * (b * c)$. Now,

(3)Hence $(a * b) * c = a * (b * c)$ and the operation is associative.