Binary operations of addition and multiplication. Prove that the set $\mathbb{R}$ of all real numbers is an abelian group under the binary operation of addition (+) and a semi-group under multiplication ($\cdot$).
Prove that $\mathbb{R}^*$, the subset of non-zero $x \in \mathbb{R}$, forms an abelian group under multiplication.
 A: The point is, that the definition of semigroup, group, Abelian group are all originated from the addition and multiplication of numbers, primarily.
So to speak, this must be a trivial exercise. You have to agree with the following:


*

*If $u,v\in\Bbb R$, then both symbols $u+v$ and $u\cdot v$ are uniquely defined in this context, and they are still elements of $\Bbb R$.

*For any elements $u,v,w\in\Bbb R$, both the equations
$$u+(v+w)=(u+v)+w, \quad\quad u\cdot(v\cdot w)=(u\cdot v)\cdot w$$
hold (associativity - it basically means we can live without parenthesis if using only $+$ or only $\cdot$ operation).

*Abelianness: For any elements $u,v\in\Bbb R$, we have both
$$u+v=v+u,\quad\quad u\cdot v=v\cdot u\,.$$
This is called commutativity, and together with associativity means that we can arbitrarily permute the members of a sum/product, and will still get the same result.

*For $\Bbb R^*$ being group, we need that the product of elements of $\Bbb R^*$ stays within $\Bbb R^*$, and that all $u\in\Bbb R^*$ has an inverse with respect to the given operation, which is now the multiplication. So, it speaks about that we can form $1/u$ whenever $u\in\Bbb R^*$.

