- Addition: for each and there exists a real number
- Multiplication: for each and there exists a real number
- Addition and Multiplication Together: for all , we have the law
- Avoiding collapse: we assume
has an order relation < to classify each number as positive, negative, or zero, based on their relation to 0. These properties interact with and to manipulate inequalities.
The implication that there are no missing points in the real number line, unlike the rational set which excludes instances of irrational numbers.
- Only universal quantifiers, for all:
- Contain an existential quantifier, exists, to prove the existence of something:
A structure is a ring if is a non-empty set and and and binary operations:
- zero element:
Addition and Multiplication:
False can imply true, but true will never imply false.