Problem
Show that $(x + y) + z = x + (y + z)$ for all $x,y,z \in \mathbb{F}^n$
Solution
Since addition in $\mathbb{F}^n$ is defined coordinate-wise, the result follows from the result shown in Problem 5 as such: $$(x + y) + z \stackrel{D1.12}{=} (x_1 + y_1, x_2 + y_2, \dots, x_n + y_n) + (z_1, z_2, \dots, z_n)$$ $$\stackrel{D1.12}{=} ((x_1 + y_1) + z_1, (x_2 + y_2) + z_2, \dots, (x_n + y_n) + z_n)$$ $$= (x_1 + (y_1 + z_1), x_2 + (y_2 + z_2), \dots, x_n + (y_n + z_n))$$ $$\stackrel{D1.12}{=} (x_1, x_2, \dots, x_n) + (y_1 + z_1, y_2 + z_2, \dots, y_n + z_n) = x + (y + z)$$ Thus addition is associative in $\mathbb{F}^n$.