Since I'm learning Math in a non-mother language I have some difficulty understanding a concept even if I use English/Arabic.
Problem: Understanding Groups in Algebraic Structure. Way to help me understand: "Soit $E$ un ensemble non vide, on appelle loi de composition interne sur $E$ toute application $\Psi$ de $E^2$ dans $E$. On note le resultat de l'operation $\Psi$ de $x$ par $y$ par un symbole, par example $\Psi(x,y) = x ⊕ y$"
Questions: What is the use of $\Psi$ in that position '$\Psi(x,y)$' and is "$Ψ(x,y) = (G, +)$" ($G$ and $+$ are the most used symbols in the tutorials in youtube)
Welcome.
$\Psi(x,y)=(G,+)$ is nonsense; I am certain the authors did not write that. You could define a group $G$ as a triplet $(G,\Psi;e)$ though. However:
Really, a group "is" (can be defined as) a nonempty set with a function $\Psi:G\times G\to G$ satisfying certain assumptions. This function is sometimes called "the group law" or "the group multiplication". However, standard function notation isn't very human-friendly here: it's really cumbersome to work with expressions like $\Psi(x,\Psi(\Psi(y,z),a))$ but $x+y+z+a$ is great! So we prefer to use a simple symbol such as "$+$" or $\cdot$ (or just no symbol at all!) to denote the 'action' of $\Psi$. This is what your text is saying: instead of using $\Psi(x,y)$, it's convenient to rewrite this as $x+y$ or $x\oplus y$ .. whatever. The symbol used doesn't matter mathematically (but it does if you want your conventions to match those of other people). For example, "$+$" makes me think the group is Abelian, whereas "$\cdot$" (so $\Psi(x,y)=x\cdot y$) makes me think of a more general, possibly anabelian group.