Consider the following definition for a binary operation on the real numbers.
Let $x$ and $y$ be real numbers. Define the operation $*$ as follows.
$$x * y = \begin{cases} x, & \text{if} & x \geq y \\ y, & \text{if} & x \leq y. \end{cases}$$
Let $a$, $b$ and $c$ be real numbers. Prove that $(a * b) * c = a * (b * c).$
As a first attempt, we could use the Trichotomy Law and treat each case separately. Although, when we think of the number of cases that we have to consider, we easily give up on this approach. (Since we would have to consider about $13$ cases, if I’m not wrong.)
I am seeking now an alternative (and possibly, a more efficient) proof for this result.
Does anyone have an idea of how to do it?
I also tried to rely on the meaning of this operation, i.e., looking at the definition, we easily conclude that $a * b$ gives the greatest number, if $a \neq b$, or just $a$ or $b$ in the case $a = b.$ But this not seems very formal to me.
As you say, this is precisely the maximum operation $\text{max}(x, y)$, and $\text{max}$ is associative because
$$\text{max}(\text{max}(x, y), z) = \text{max}(x, \text{max}(y, z)) = \text{max}(x, y, z)$$
is just the maximum among the three numbers $x, y, z$; no casework is necessary.