Showing ((a1 ⋄ a2) ⋄ a3) ⋄ a4 = a1 ⋄ (a2 ⋄ (a3 ⋄ a4)) in a semigroup.

Question

I am a student in computer science - first year. I study linear linear algebra 2 - course of linear algebra 1. - In some institutions academic studies teach the courses together / teach in another way.

I tried to solve the question a few hours but I'm not sure how to solve it exactly.

"4. (somewhat harder) Let G4 = (A, ⋄) be a semi-group, with |A| ≥ 4. Prove that ∀a1, a2, a3, a4 ∈ G4 : ((a1 ⋄ a2) ⋄ a3) ⋄ a4 = a1 ⋄ (a2 ⋄ (a3 ⋄ a4))."

There are two data in the question - a semi-group, and that A is an absolute value .greater than 4. how A > 4 helps me solve the question, I did not understand how to use it at all. What does it actually help me?

Answer 1

$$((a*b)*c)*d = (a*(b*c))*d = a*((b*c)*d) = a*(b*(c*d))$$

Answer 2

We have

\begin{align} \color{blue}{(}(a_1\diamond a_2)\diamond a_3\color{blue}{)}\diamond a_4&=\color{blue}{(}a_1\diamond \color{red}{(a_2\diamond a_3)}\color{blue}{)}\diamond a_4 \\ &=a_1\diamond \color{green}{(}\color{red}{(a_2\diamond a_3)}\diamond a_4\color{green}{)} \\ &=a_1\diamond \color{green}{(}a_2\diamond (a_3\diamond a_4)\color{green}{)}. \end{align}