Addition and Multiplication.

We recall the main properties of addition and multiplication in $\mathbb{R}$ .

1. Addition:
   • Closure:

     $$\forall x \in \mathbb{R}, \forall y \in \mathbb{R}, x+y \in \mathbb{R}.$$

     
     

   • Associativity:

     $$\forall x \in \mathbb{R}, \forall y \in \mathbb{R}, \forall z \in \mathbb{R}, (x+y)+z=x+(y+z).$$

     
     

   • Commutativity:

     $$\forall x \in \mathbb{R}, \forall y \in \mathbb{R}, x+y=y+x.$$

     
     

   • Neutral element:

     $$\forall x \in \mathbb{R}, x+0=0+x=x.$$

     
     

   • Opposite element:

     $$\forall x \in \mathbb{R}, x+(-x)=(-x)+x=0.$$

     
     

2. Multiplication:
   • Closure:

     $$\forall x \in \mathbb{R}, \forall y \in \mathbb{R}, x \cdot y \in \mathbb{R}.$$

     
     

   • Associativity:

     $$\forall x \in \mathbb{R}, \forall y \in \mathbb{R}, \forall z \in \mathbb{R}, (x \cdot y) \cdot z=x \cdot (y \cdot z).$$

     
     

   • Commutativity:

     $$\forall x \in \mathbb{R}, \forall y \in \mathbb{R}, xy=yx.$$

     
     

   • Neutral element:

     $$\forall x \in \mathbb{R}, x \cdot 1 =1 \cdot x=x.$$

     
     

   • Opposite element:

     $$\forall x \in \mathbb{R}-{0}, x \cdot \frac 1x =\frac 1x \cdot x=1.$$

     
     

3. Distributivity:

   $$\forall x \in \mathbb{R}, \forall y \in \mathbb{R}, \forall z \in \mathbb{R}, x(y+z)=xy+xz \; \; (x+y)z=xz+yz.$$