The Real Number Axioms

The set of real numbers, $\mathbb{R}$, satisfies the following three sets of axioms.

The Field Axioms

  1. Closure Under Addition : $\forall$ x,y $\in$ $\mathbb{R},$ $\exists$ an entity called x+y which is also an element of $\mathbb{R}$.
  2. Associative Law Under Addition : $\forall$x,y,z $\in \mathbb{R},$ (x + y) + z = x + (y + z).
  3. Commutative Law Under Addition : $\forall$ x,y $\in \mathbb{R},$ x + y = y + x.
  4. Additive Identity : $\exists$ an entity denoted as '0' such that $\forall \in \mathbb{R},$ x + 0 = 0 + x = x.
  5. Additive Inverse : $\forall x \in \mathbb{R}, \exists $ a y $\in \mathbb{R}$ such that x + y = y + x = 0.
  6. Closure Under Multiplication : $\forall$ x,y $\exists$ an entity called x $\cdot$ y which is also an element of $\mathbb{R}$.
  7. Associative Law Under Multiplication : $\forall$ x,y,z $\in \mathbb{R}, (x \cdot y) \cdot z = x \cdot (y \cdot z).$
  8. Commutative Law Under Multiplication : $\forall x,y \in \mathbb{R}, x \cdot y = y \cdot x.$
  9. Multiplicative Identity : $\exists$ an entity denoted as '1' such that $\forall$ x $\in \mathbb{R}, x \cdot 1 = 1 \cdot x = x.$
  10. Multiplicative Inverse : $\forall x \in \mathbb{R}, \exists$ a $y \in \mathbb{R}$ such that $x \cdot y = y \cdot x = 1.$
  11. Distributive Law Under Multiplication : $\forall$ x, y, z $\in \mathbb{R}, x \cdot (y + z) = x \cdot y + x \cdot z.$

Order Axioms

  1. If x,y $\in \mathbb{R^+}$ then x+y and x $\cdot$ y are also elements of $\mathbb{R^+}.$
  2. For every x $\in \mathbb{R}$ only one of the following three conditions holds:
    1. x $\in \mathbb{R}^+.$
    2. x = 0.
    3. -x $\in \mathbb{R}^+.$

Completeness Axiom

  1. Least Upper Bound : Let $A$ be a set of real numbers that has a upper bound1. $x_{{\text{lub}}}$ $\in \mathbb{R}$ is called a least upper bound of $A$ if:
    1. $x_{{\text{lub}}}$ is an upper bound of $A$.
    2. $\forall$ upper bounds, $x$, which are elements of $A$, $x_{{\text{lub}}} \leq x.$

1Upper Bound
$x_{{\text{ub}}}$ is an upper bound for set $A$ if $x_{{\text{ub}}} \in \mathbb{R}$ such that $\forall a \in A$, $x_{{\text{ub}}} > a.$
Boundedness
A set $A$ of real numbers is called bounded from above is $A$ has at least one upper bound.
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